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Device driver

In the context of an operating system, a device driver is a computer program that operates or controls a particular type of device that is attached to a computer or automaton. A driver provides a software interface to hardware devices, enabling operating systems and other computer programs to access hardware functions without needing to know precise details about the hardware being used.

A driver communicates with the device through the computer bus or communications subsystem to which the hardware connects. When a calling program invokes a routine in the driver, the driver issues commands to the device (drives it). Once the device sends data back to the driver, the driver may invoke routines in the original calling program.

Drivers are hardware dependent and operating-system-specific. They usually provide the interrupt handling required for any necessary asynchronous time-dependent hardware interface.

Purpose
The main purpose of device drivers is to provide abstraction by acting as a translator between a hardware device and the applications or operating systems that use it. Programmers can write higher-level application code independently of whatever specific hardware the end-user is using. For example, a high-level application for interacting with a serial port may simply have two functions for "send data" and "receive data". At a lower level, a device driver implementing these functions would communicate to the particular serial port controller installed on a user's computer. The commands needed to control a 16550 UART are much different from the commands needed to control an FTDI serial port converter, but each hardware-specific device driver abstracts these details into the same (or similar) software interface.

Development
Writing a device driver requires an in-depth understanding of how the hardware and the software works for a given platform function. Because drivers require low-level access to hardware functions in order to operate, drivers typically operate in a highly privileged environment and can cause system operational issues if something goes wrong. In contrast, most user-level software on modern operating systems can be stopped without greatly affecting the rest of the system. Even drivers executing in user mode can crash a system if the device is erroneously programmed. These factors make it more difficult and dangerous to diagnose problems.

The task of writing drivers thus usually falls to software engineers or computer engineers who work for hardware-development companies. This is because they have better information than most outsiders about the design of their hardware. Moreover, it was traditionally considered in the hardware manufacturer's interest to guarantee that their clients can use their hardware in an optimum way. Typically, the Logical Device Driver (LDD) is written by the operating system vendor, while the Physical Device Driver (PDD) is implemented by the device vendor. However, in recent years, non-vendors have written numerous device drivers for proprietary devices, mainly for use with free and open source operating systems. In such cases, it is important that the hardware manufacturer provide information on how the device communicates. Although this information can instead be learned by reverse engineering, this is much more difficult with hardware than it is with software.

Microsoft has attempted to reduce system instability due to poorly written device drivers by creating a new framework for driver development, called Windows Driver Frameworks (WDF). This includes User-Mode Driver Framework (UMDF) that encourages development of certain types of drivers—primarily those that implement a message-based protocol for communicating with their devices—as user-mode drivers. If such drivers malfunction, they do not cause system instability. The Kernel-Mode Driver Framework (KMDF) model continues to allow development of kernel-mode device drivers, but attempts to provide standard implementations of functions that are known to cause problems, including cancellation of I/O operations, power management, and plug and play device support.

Apple has an open-source framework for developing drivers on macOS, called I/O Kit.

In Linux environments, programmers can build device drivers as parts of the kernel, separately as loadable modules, or as user-mode drivers (for certain types of devices where kernel interfaces exist, such as for USB devices). Makedev includes a list of the devices in Linux, including ttyS (terminal), lp (parallel port), hd (disk), loop, and sound (these include mixer, sequencer, dsp, and audio).

Microsoft Windows .sys files and Linux .ko files can contain loadable device drivers. The advantage of loadable device drivers is that they can be loaded only when necessary and then unloaded, thus saving kernel memory.

Privilege levels
Depending on the operating system, device drivers may be permitted to run at various different privilege levels. The choice of which level of privilege the drivers are in is largely decided by the type of kernel an operating system uses. An operating system which uses a monolithic kernel, such as the Linux kernel, will typically run device drivers with the same privilege as all other kernel objects. By contrast, a system designed around microkernel, such as Minix, will place drivers as processes independent from the kernel but that use it for essential input-output functionalities and to pass messages between user programs and each other. On Windows NT, a system with a hybrid kernel, it is common for device drivers to run in either kernel-mode or user-mode.

The most common mechanism for segregating memory into various privilege levels is via protection rings. On many systems, such as those with x86 and ARM processors, switching between rings imposes a performance penalty, a factor that operating system developers and embedded software engineers consider when creating drivers for devices which are preferred to be run with low latency, such as network interface cards. The primary benefit of running a driver in user mode is improved stability, since a poorly written user-mode device driver cannot crash the system by overwriting kernel memory.

Applications
Because of the diversity of hardware and operating systems, drivers operate in many different environments. Drivers may interface with:

 * Printers
 * Video adapters
 * Network cards
 * Sound cards
 * PC chipsets
 * Local buses of various sorts—in particular, for bus mastering on modern systems
 * Low-bandwidth I/O buses of various sorts (for pointing devices such as mice, keyboards, etc.)
 * Computer storage devices such as hard disk, CD-ROM, and floppy disk buses (ATA, SATA, SCSI, SAS)
 * Implementing support for different file systems
 * Image scanners
 * Digital cameras
 * Digital terrestrial television tuners
 * Radio frequency communication transceiver adapters for wireless personal area networks as used for short-distance and low-rate wireless communication in home automation, (such as example Bluetooth Low Energy (BLE), Thread, Zigbee, and Z-Wave).
 * IrDA adapters

Common levels of abstraction for device drivers include:
 * For hardware:
 * Interfacing directly
 * Writing to or reading from a device control register
 * Using some higher-level interface (e.g. Video BIOS)
 * Using another lower-level device driver (e.g. file system drivers using disk drivers)
 * Simulating work with hardware, while doing something entirely different
 * For software:
 * Allowing the operating system direct access to hardware resources
 * Implementing only primitives
 * Implementing an interface for non-driver software (e.g. TWAIN)
 * Implementing a language, sometimes quite high-level (e.g. PostScript)

So choosing and installing the correct device drivers for given hardware is often a key component of computer system configuration.

Virtual device drivers
Virtual device drivers represent a particular variant of device drivers. They are used to emulate a hardware device, particularly in virtualization environments, for example when a DOS program is run on a Microsoft Windows computer or when a guest operating system is run on, for example, a Xen host. Instead of enabling the guest operating system to dialog with hardware, virtual device drivers take the opposite role and emulates a piece of hardware, so that the guest operating system and its drivers running inside a virtual machine can have the illusion of accessing real hardware. Attempts by the guest operating system to access the hardware are routed to the virtual device driver in the host operating system as e.g., function calls. The virtual device driver can also send simulated processor-level events like interrupts into the virtual machine.

Virtual devices may also operate in a non-virtualized environment. For example, a virtual network adapter is used with a virtual private network, while a virtual disk device is used with iSCSI. A good example for virtual device drivers can be Daemon Tools.

There are several variants of virtual device drivers, such as VxDs, VLMs, and VDDs.

Open source drivers

 * Graphics device driver
 * Printers: CUPS
 * RAIDs: CCISS (Compaq Command Interface for SCSI-3 Support )
 * Scanners: SANE
 * Video: Vidix, Direct Rendering Infrastructure

Solaris descriptions of commonly used device drivers:
 * fas: Fast/wide SCSI controller
 * hme: Fast (10/100 Mbit/s) Ethernet
 * isp: Differential SCSI controllers and the SunSwift card
 * glm: (Gigabaud Link Module ) UltraSCSI controllers
 * scsi: Small Computer Serial Interface (SCSI) devices
 * sf: soc+ or social Fiber Channel Arbitrated Loop (FCAL)
 * soc: SPARC Storage Array (SSA) controllers and the control device
 * social: Serial optical controllers for FCAL (soc+)

APIs

 * Windows Display Driver Model (WDDM) – the graphic display driver architecture for Windows Vista and later.
 * Unified Audio Model (UAM)
 * Windows Driver Foundation (WDF)
 * Declarative Componentized Hardware (DCH) - Universal Windows Platform driver
 * Windows Driver Model (WDM)
 * Network Driver Interface Specification (NDIS) – a standard network card driver API
 * Advanced Linux Sound Architecture (ALSA) – the standard Linux sound-driver interface
 * Scanner Access Now Easy (SANE) – a public-domain interface to raster-image scanner-hardware
 * Installable File System (IFS) – a filesystem API for IBM OS/2 and Microsoft Windows NT
 * Open Data-Link Interface (ODI) – network card API similar to NDIS
 * Uniform Driver Interface (UDI) – a cross-platform driver interface project
 * Dynax Driver Framework (dxd) – C++ open source cross-platform driver framework for KMDF and IOKit

Identifiers
A device on the PCI bus or USB is identified by two IDs which consist of two bytes each. The vendor ID identifies the vendor of the device. The device ID identifies a specific device from that manufacturer/vendor.

A PCI device has often an ID pair for the main chip of the device, and also a subsystem ID pair which identifies the vendor, which may be different from the chip manufacturer.

Security
Devices often have a large number of diverse and customized device drivers running in their operating system (OS) kernel and often contain various bugs and vulnerabilities, making them a target for exploits. Bring Your Own Vulnerable Driver (BYOVD) uses signed, old drivers that contain flaws that allow hackers to insert malicious code into the kernel.

There is a lack of effective kernel vulnerability detection tools, especially for closed-source OSes such as Microsoft Windows where the source code of the device drivers is mostly not public (open source) and the drivers often also have many privileges.

Such vulnerabilities also exist in drivers in laptops, drivers for WiFi and bluetooth, gaming/graphics drivers, and drivers in printers.

A group of security researchers considers the lack of isolation as one of the main factors undermining kernel security, and published a isolation framework to protect operating system kernels, primarily the monolithic Linux kernel which, according to them, gets ~80,000 commits/year to its drivers.

title: Device driver


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Dimona

Dimona (דִּימוֹנָה, ديمونا) is an Israeli city in the Negev desert, 30 km to the south-east of Beersheba and 35 km west of the Dead Sea above the Arava valley in the Southern District of Israel. In its population was. The Shimon Peres Negev Nuclear Research Center, colloquially known as the Dimona Reactor, is located 13 km southeast of the city.

Etymology
The Negev Naming Committee chose the name based upon that of a biblical town, mentioned in Joshua 15:21-22, on the basis that "the sound of this name had been preserved in the Arabic name Harabat Umm Dumna."

History

Dimona was one of the development towns created in the 1950s under the leadership of Israel's first Prime Minister, David Ben-Gurion. Dimona itself was conceived in 1953. The location chosen was close to the Dead Sea Works. It was established in 1955. The first residents were Jewish immigrants from North Africa, with an initial 36 families being the first to settle there. Its population in 1955 was about 300. The North African immigrants also constructed the city's houses. The population was composed mainly of North African, particularly Moroccan immigrants, though immigrants from Yemen and Eastern Europe also arrived, as did Bene Israel immigrants from India.

When the Israeli nuclear program began in 1958, a location not far from the city was chosen for the Negev Nuclear Research Center due to its relative isolation in the desert and availability of housing. In the late 1950s and early 1960s, immigrants from Eastern Europe arrived. A textile factory was opened in 1958. That same year, Dimona became a local council. In 1961, it had a population of 5,000. The emblem of Dimona (as a local council), adopted 2 March 1961, appeared on a stamp issued on 24 March 1965. Dimona was declared a city in 1969. In 1971, it had a population of 23,700.

In spite of a gradual decrease during the 1980s, the city's population began to grow once again in the 1990s when it took in immigrants from the former Soviet Union and Ethiopia. Currently, Dimona is the third largest city in the Negev, with the population of almost 34,000. Due to projected rapid population growth in the Negev, the city is expected to triple in size by 2025.

Demography
Dimona is described as "mini-India" by many for its 7,500-strong Indian Jewish community. It is also home to Israel's Black Hebrew community, formerly governed by its founder and spiritual leader, Ben Ammi Ben-Israel, now deceased. The Black Hebrews number about 3,000 in Dimona, with additional families in Arad, Mitzpe Ramon and the Tiberias area. Their official status in Israel was an ongoing issue for many years, but in May 1990, the issue was resolved with the issuing of first B/1 visas, and a year later, issuing of temporary residency. Status was extended to August 2003, when the Israeli Ministry of Interior granted permanent residency.

Economy
In the early 1980s, textile plants, such as Dimona Textiles Ltd., dominated the industrial landscape. Many plants have since closed. Dimona Silica Industries Ltd. manufactures precipitated silica and calcium carbonate fillers. About a third of the city's population works in industrial workplaces (chemical plants near the Dead Sea like the Dead Sea Works, high-tech companies and textile shops), and another third in the area of services. Due to the introduction of new technologies, many workers have been made redundant in the recent years, creating a total unemployment rate of about 10%. Dimona has taken part of Israel's solar transformation. The Rotem Industrial Complex outside of the city has dozens of solar mirrors that focus the sun's rays on a tower that in turn heats a water boiler to create steam, turning a turbine to create electricity. Luz II, Ltd. plans to use the solar array to test new technology for the three new solar plants to be built in California for Pacific Gas and Electric Company.

Geography and climate
Dimona is located in the Negev Desert. The city stands at an elevation of around 550 - 600 m above sea level.

Climate
Dimona has a semi-arid climate (Köppen climate classification: BSh). The average annual temperature is 18.5 °C, and around 213 mm of precipitation falls annually.

Transportation
In the early 1950s, an extension to Dimona and south was constructed from the Railway to Beersheba, designed for freight traffic. A passenger service began in 2005, after pressure from Dimona's municipality. Dimona Railway Station is located in the southwestern part of the city. The main bus terminal is the Dimona Central Bus Station, with lines to Beersheba, Tel Aviv, Eilat, and nearby towns.

Notable people

 * Lucy Aharish, News presenter, Reporter, Television host
 * Jacques Amir (1933–2011), Knesset member
 * Yossi Benayoun (born 1980), Footballer
 * Meir Cohen (born 1955), Minister of Welfare and Social Services, Knesset member
 * Kfir Edri (born 1976), Footballer
 * Ben Israel Ben Ammi (1939–2014), Spiritual leader of the African Hebrew Israelites of Jerusalem (Shomrei Hashalom)

Twin towns
Dimona is twinned with:
 * 🇩🇪 Andernach, Rhineland-Palatinate, Germany (1975)

license: Creative Commons - Attribution Share-Alike - https://creativecommons.org/licenses/by-sa/4.0/

title: Dimona


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Talk:Dimona

The discussion between Uriyan, SJK and other has been moved to user talk:Uriyan

policy on placing Dimona images here?
There are a few images of Dimona, some by Mordechai Vanunu, others by spy satellites, that I'd like to use to illustrate what it looks like.

There is no problem from the website owner, I asked her permission and here's the text of her reply; I've munged the email addresses on request for anti-spam reasons:

Date: Wed, 28 Apr 2004 08:29:38 -0700 (GMT-07:00) From: To:   Subject: Re: nuclearweaponarchive.org - permission to use images of Dimona?

It's all right with *me*, but I claim no rights to those images.

I am of the opinion that it would be impossible for anyone to claim rights to the Vanunu photograph, except maybe for the Israeli government in whose facility the photo was illegally obtained. But no one legal body seems interested in enforcing that right, including the Israeli government itself. So it I think it is clearly in the public doman.

-Original Message- From:  Sent: Apr 22, 2004 1:10 PM To: Subject: nuclearweaponarchive.org - permission to use images of Dimona? Dear , I found your email address on the nuclearweaponarchive.org website and would like permission to use 2 images from the Israel section for use in the Wikipedia article on Dimona. Wikipedia can only use public domain (or GFDL) material. The two images I have in mind are:

The two images I have in mind are: http://nuclearweaponarchive.org/Israel/Dimona_compare.jpg and http://nuclearweaponarchive.org/Israel/Dimona1s.jpg (courtesy Mordechai Vanunu)

What do other Wikipedians think? -Wikibob | Talk 19:58, 2004 Apr 28 (UTC)

God bless BBC!
BBC Newsnight yesterday revealed everything about UK's sale-out of much nuclear goodies towards the 1960's zionist bomb. Nobody else dares to ask Vanunu nowadays for fear of the Mossad but BBC. BBC quotes Vanunu Israel made lithium H-bombs not just A-bombs. Of course only Iran gets invaded ... the zionists are untouchable.

Read it all here: http://news.bbc.co.uk/1/hi/programmes/newsnight/4789832.stm and here http://news.bbc.co.uk/1/hi/programmes/newsnight/4743493.stm and here http://news.bbc.co.uk/1/shared/bsp/hi/pdfs/israelbomb.pdf

Relevancy?
Maybe its just me, but this article talks awfully lot about Israel's nuclear programme, though its supposed to be on the city of Dimona. Bless sins 02:30, 12 March 2006 (UTC)
 * Just as the Windscale article barely bothers to mention the town. Possibly the reactor needs its own page. Rsynnott 16:06, 4 December 2006 (UTC)

Seperate into 2 pages - Dimona and Negev Nuclear Research Center
Why does this article include the nuclear research center? I think it should be divided into two articles: Dimona and Negev Nuclear Research Center, especially given that the Nuclear center isn't in Dimona but rather a few miles away. Any objections? Eranb 12:05, 16 May 2006 (UTC)

I object, the nuclear research center is the thing dimona is most known for. thats what i tihnk of when i hear the name.160.39.168.58 09:14, 12 December 2006 (UTC)

population
the article and the info box report different pop numbers... someone should check that —The preceding unsigned comment was added by 160.39.168.58 (talk) 09:13, 12 December 2006 (UTC).

Last Paragraph
The last paragraph reads like a plug for Black Hebrews, and was written (poorly) by someone who either belongs to or sympathizes with the movement. Either rewrite the last paragraph so that it is NPOV, or I will delete it. Narsil27 22:39, 14 March 2007 (UTC)

Coordinate error
The following coordinate fixes are needed for lat 31.002447, lon 35.146493

—194.60.90.4 (talk) 14:26, 8 August 2012 (UTC)
 * ❌. The coordinates currently in the article are the correct ones for the settlement of Dimona. The coordinates you've given are those of the Negev Nuclear Research Center. Deor (talk) 22:59, 8 August 2012 (UTC)

Rafa?
added this to the paragraph about the Nuclear Research Center on Feb 10, 2015: (diff):

 * However, Israel's nuclear storage facility has been transferred to Rafa (a change which, mysteriously, does not appear on Google Earth) - due to the increasing number of Alawites among the Bedouin tribesmen who have considerable experience with explosives.

The parenthetical remark sounds more like a polemic than something you might expect in an encyclopedia. But the whole sentence does not make much sense as long as we don't know which "Rafa" is meant here. The border town Rafah in the Gaza strip sounds like an implausible choice for storage of nuclear material, but what other place could be meant? Also, no sources are given, so I deleted the sentence.

--Austrian (talk) 22:15, 18 February 2015 (UTC)

External links modified
Hello fellow Wikipedians,

I have just modified one external link on Dimona. Please take a moment to review my edit. If you have any questions, or need the bot to ignore the links, or the page altogether, please visit this simple FaQ for additional information. I made the following changes:
 * Added archive https://web.archive.org/web/20050108103359/http://www.blink.org.uk/pdescription.asp?key=2888&grp=18 to http://www.blink.org.uk/pdescription.asp?key=2888&grp=18

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title: Talk:Dimona


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DC Comics

DC Comics, Inc. (doing business as DC) is an American comic book publisher and the flagship unit of DC Entertainment, a subsidiary of Warner Bros. Discovery. DC is an initialism for "Detective Comics", an American comic book series first published in 1937.

DC Comics is one of the largest and oldest American comic book companies, with their first comic under the DC banner being published in 1937. The majority of its publications take place within the fictional DC Universe and feature numerous culturally iconic heroic characters, such as Superman, Batman, Wonder Woman, Green Lantern, the Flash, and Aquaman; as well as famous fictional teams including the Justice League, the Justice Society of America, the Teen Titans, and the Suicide Squad. The universe also features an assortment of well-known supervillains such as the Joker, Lex Luthor, Deathstroke, the Reverse-Flash, Brainiac, and Darkseid. The company has published non-DC Universe-related material, including Watchmen, V for Vendetta, Fables and many titles under their alternative imprint Vertigo and now DC Black Label.

Originally in Manhattan at 432 Fourth Avenue, the DC Comics offices have been located at 480 and later 575 Lexington Avenue; 909 Third Avenue; 75 Rockefeller Plaza; 666 Fifth Avenue; and 1325 Avenue of the Americas. DC had its headquarters at 1700 Broadway, Midtown Manhattan, New York City, but DC Entertainment relocated its headquarters to Burbank, California in April 2015.

Penguin Random House Publisher Services distributes DC Comics' books to the bookstore market, while Diamond Comic Distributors supplied the comics shop direct market until June 2020, when Lunar Distribution and UCS Comic Distributors, who already dominated direct market distribution on account of the disruption to Diamond that resulted from the COVID-19 pandemic, replaced Diamond to distribute to that market.

DC Comics and its longtime major competitor Marvel Comics (acquired in 2009 by The Walt Disney Company, Warner Bros. Discovery's main competitor) together shared approximately 70% of the American comic book market in 2017, though this number may give a distorted view since graphic novels are excluded. With the sales of all books included, DC is the second biggest publisher, after Viz Media, and Marvel is third.

Golden Age

Entrepreneur Major Malcolm Wheeler-Nicholson founded National Allied Publications in 1934 intended as an American comic book publishing company. The first publishing of the company debuted with the tabloid-sized New Fun: The Big Comic Magazine #1 (the first of a comic series later called More Fun Comics) with a cover date of February 1935. It was an anthology title essentially for original stories not reprinted from newspaper strips, unlike many comic book series before it. While superhero comics are what DC Comics is known for throughout modern times, the genres in the first anthology titles consisted of funnies, Western comics and adventure-related stories. The character Doctor Occult, created by Jerry Siegel and Joe Shuster in December 1935 with issue No. 6 of New Fun Comics, is considered the earliest recurring superhero created by DC who is still used. The company created a second recurring title called New Comics No. 1, released in December 1935, which was the start of the long-running Adventure Comics series featuring many anthology titles as well. Shortly afterwards, the group became Nicholson Publishing by 1936.

Wheeler-Nicholson's next and final title, Detective Comics, advertised with a cover illustration dated December 1936, eventually premiered three months late with a March 1937 cover date. The themed anthology that revolved originally around fictional detective stories became in modern times the longest-running ongoing comic series. A notable debut in the first issue was Slam Bradley, created in a collaboration between Malcolm Wheeler-Nicholson, Jerry Siegel and Joe Shuster. In 1937, in debt to printing-plant owner and magazine distributor Harry Donenfeld — who also published pulp magazines and operated as a principal in the magazine distributorship Independent News — Wheeler-Nicholson had to take Donenfeld on as a partner to publish Detective Comics No. 1. Detective Comics, Inc. (which would help inspire the abbreviation DC) was formed, with Wheeler-Nicholson and Jack S. Liebowitz, Donenfeld's accountant, listed as owners. Major Wheeler-Nicholson remained for a year, but cash-flow problems continued, and he was forced out. Shortly afterwards, Detective Comics, Inc. purchased the remains of National Allied, also known as Nicholson Publishing, at a bankruptcy auction and absorbed it.

Meanwhile, Max Gaines formed the sister company All-American Publications in 1939. Detective Comics, Inc. soon launched a new anthology title, entitled Action Comics. Issue#1, cover dated June 1938, first featured characters such as Superman by Siegel and Shuster, Zatara by Fred Guardineer and Tex Thompson by Ken Finch and Bernard Baily. It is considered to be the first comic book to feature the new character archetype, soon known as "superheroes", and was a sales hit bringing to life a new age of comic books, with the credit going to the first appearance of Superman both being featured on the cover and within the issue. It is now one of the most expensive and valuable comic book issues of all time. The issue's first featured tale which starred Superman was the first to feature an origin story of superheroes with the reveal of an unnamed planet, later known as Krypton, that he is said to be from. The issue also contained the first essential supporting character and one of the earliest essential female characters in comics with Lois Lane as Superman's first depicted romantic interest. The Green Hornet-inspired character known as the Crimson Avenger by Jim Chamber was featured in Detective Comics No. 20 (October 1938). The character makes a distinction of being the first masked vigilante published by DC. An unnamed "office boy" retconned as Jimmy Olsen's first appearance was revealed in Action Comics #6's (November 1938) Superman story by Siegel and Shuster.

Starting in 1939, Siegel and Shuster's Superman would be the first comic-derived character to appear outside of comic magazines and later appear in newspaper strips starring himself, which first introduced Superman's biological parents, Jor-El and Lara. All-American Publications' first comic series called All-American Comics was first published in April 1939. The series Detective Comics would make successful history as first featuring Batman by Bob Kane and Bill Finger in issue No.27 (March 1939) with the request of more superhero titles. Batman was depicted as a masked vigilante wearing a caped suit known as the Batsuit, along with riding a car that would later be referred to as the Batmobile. Also within the Batman story was the supporting character, James Gordon, Police commissioner of what later would be Gotham City Police Department. Despite being a parody, All-American Publications introduced the earliest female character who would later be a female superhero called Red Tornado (though disguised as a male) in Ma Hunkel who first appeared in the "Scribbly" stories in All-American Comics No. 3 (June 1939). Another important Batman debut was the introduction of the fictional mansion known as Wayne Manor first seen in Detective Comics No. 28 (June 1939). The series Adventure Comics would eventually follow in the footsteps of Action Comics and Detective Comics, featuring a new recurring superhero. The superhero called Sandman was first written in issue No. 40 (cover date: July 1939). Action Comics No. 13 (June 1939) introduced the first recurring Superman enemy referred to as the Ultra-Humanite first introduced by Siegel and Shuster, commonly cited as one of the earliest supervillains in comic books. The character Superman had another breakthrough when he was given his own comic book, which was unheard of at the time. The first issue, introduced in June 1939, helped directly introduce Superman's adoptive parents, Jonathan and Martha Kent, by Siegel and Shuster. Detective Comics #29 (July 1939) introduced the Batman's utility belt by Gardner Fox. Outside of DC's publishing, a character later integrated as DC was introduced by Fox Feature Syndicate named the Blue Beetle released in August 1939. Fictional cities would be a common theme of DC. The first revealed city was Superman's home city, Metropolis, that was originally named in Action Comics No. 16 in September 1939. Detective Comics No. 31 in September 1939 by Gardner Fox, Bob Kane and Sheldon Moldoff introduced a romantic interest of Batman named Julie Madison, the weapon known as the Batarang that Batman commonly uses, and the fictional aircraft called the Batplane. Batman's origin would first be shown in Detective Comics No. 33 (Nov. 1939) first depicting the death of Thomas Wayne and Martha Wayne by a mugger. The origin story would remain crucial for the fictional character since the inception. The Daily Planet (a common setting of Superman) was first named in a Superman newspaper strip around November 1939. The superhero Doll Man was the first superhero by Quality, which DC now owns. Fawcett Comics was formed around 1939 and would become DC's original competitor company in the next decade. At the end of 1944, All-American titles now began using its own logo to distinguish it from the National comics.

All-American Publications, an affiliate concern co-owned by Max Gaines and Liebowitz, soon merged with Detective Comics, Inc., forming National Comics Publications on September 30, 1946. The year before, in June 1945, Gaines let Liebowitz buy him out, and kept only Picture Stories from the Bible as the foundation of his own new company, EC Comics. At that point, "Liebowitz promptly orchestrated the merger of All-American and Detective Comics into National Comics... Next he took charge of organizing National Comics, [the self-distributorship] Independent News, and their affiliated firms into a single corporate entity, National Periodical Publications". National Periodical Publications became publicly traded on the stock market in 1961.

Despite the official names "National Comics" and "National Periodical Publications", the company began branding itself as "Superman-DC" as early as 1940, and the company became known colloquially as DC Comics for years before the official adoption of that name in 1977.

The company began to move aggressively against what it saw as copyright-violating imitations from other companies, such as Fox Comics' Wonder Man, which (according to court testimony) Fox started as a copy of Superman. This extended to DC suing Fawcett Comics over Captain Marvel, at the time comics' top-selling character (see National Comics Publications, Inc. v. Fawcett Publications, Inc.). Faced with declining sales and the prospect of bankruptcy if it lost, Fawcett capitulated in 1953 and ceased publishing comics. Years later, Fawcett sold the rights for Captain Marvel to DC—which in 1972 revived Captain Marvel in the new title Shazam! featuring artwork by his creator, C. C. Beck. In the meantime, the abandoned trademark had been seized by Marvel Comics in 1967, with the creation of their Captain Marvel, forbidding the DC comic itself to be called that. While Captain Marvel did not recapture his old popularity, he later appeared in a Saturday morning live action TV adaptation and gained a prominent place in the mainstream continuity DC calls the DC Universe.

When the popularity of superheroes faded in the late 1940s, the company focused on such genres as science fiction, Westerns, humor, and romance. DC also published crime and horror titles, but relatively tame ones, and thus avoided the mid-1950s backlash against such comics. A handful of the most popular superhero-titles, including Action Comics and Detective Comics, the medium's two longest-running titles, continued publication.

Silver Age
In the mid-1950s, editorial director Irwin Donenfeld and publisher Liebowitz directed editor Julius Schwartz (whose roots lay in the science-fiction book market) to produce a one-shot Flash story in the try-out title Showcase. Instead of reviving the old character, Schwartz had writers Robert Kanigher and John Broome, penciler Carmine Infantino, and inker Joe Kubert create an entirely new super-speedster, updating and modernizing the Flash's civilian identity, costume, and origin with a science-fiction bent. The Flash's reimagining in Showcase No. 4 (October 1956) proved sufficiently popular that it soon led to a similar revamping of the Green Lantern character, the introduction of the modern all-star team Justice League of America (JLA), and many more superheroes, heralding what historians and fans call the Silver Age of Comic Books.

National did not reimagine its continuing characters (primarily Superman, Batman, and Wonder Woman), but radically overhauled them. The Superman family of titles, under editor Mort Weisinger, introduced such enduring characters as Supergirl, Bizarro, and Brainiac. The Batman titles, under editor Jack Schiff, introduced the successful Batwoman, Bat-Girl, Ace the Bat-Hound, and Bat-Mite in an attempt to modernize the strip with non-science-fiction elements. Schwartz, together with artist Infantino, then revitalized Batman in what the company promoted as the "New Look", with relatively down-to-Earth stories re-emphasizing Batman as a detective. Meanwhile, editor Kanigher successfully introduced a whole family of Wonder Woman characters having fantastic adventures in a mythological context.

Since the 1940s, when Superman, Batman, and many of the company's other heroes began appearing in stories together, DC's characters inhabited a shared continuity that, decades later, was dubbed the "DC Universe" by fans. With the story "Flash of Two Worlds", in Flash No. 123 (September 1961), editor Schwartz (with writer Gardner Fox and artists Infantino and Joe Giella) introduced a concept that allowed slotting the 1930s and 1940s Golden Age heroes into this continuity via the explanation that they lived on an other-dimensional "Earth 2", as opposed to the modern heroes' "Earth 1"—in the process creating the foundation for what was later called the DC Multiverse.

National Periodical Publications
DC's introduction of the reimagined superheroes did not go unnoticed by other comics companies. In 1961, with DC's JLA as the specific spur, Marvel Comics writer-editor Stan Lee and artist Jack Kirby ushered in the sub-Silver Age "Marvel Age" of comics with the debut issue of The Fantastic Four. Reportedly, DC ignored the initial success of Marvel with this editorial change until its consistently strengthening sales, albeit also benefiting Independent News' business as their distributor as well, made that impossible. That commercial situation especially applied with Marvel's superior sell-through percentage numbers which were typically 70% to DC's roughly 50%, which meant DC's publications were barely making a profit in comparison after returns from the distributors were calculated while Marvel was making an excellent profit by comparison. Also that year, both DC and Marvel became the first companies to increase prices from ten cents to twelve cents in 1961, when Dell Comics, the rival publisher cost fifteen cents.

However, the senior DC staff were reportedly at a loss at this time to understand how this small publishing house was achieving this increasingly threatening commercial strength. For instance, when Marvel's product was examined in a meeting, Marvel's emphasis on more sophisticated character-based narrative and artist-driven visual storytelling was apparently ignored for self-deluding guesses at the brand's popularity which included superficial reasons like the presence of the color red or word balloons on the cover, or that the perceived crudeness of the interior art was somehow more appealing to readers. When Lee learned about DC's subsequent experimental attempts to imitate these perceived details, he amused himself by arranging direct defiance of those assumptions in Marvel's publications as sales strengthened further to frustrate the competition.

However, this ignorance of Marvel's true appeal did not extend to some of the writing talent during this period, from which there were some attempts to emulate Marvel's narrative approach. For instance, there was the Doom Patrol series by Arnold Drake, a writer who previously warned the management of the new rival's strength; a superhero team of outsiders who resented their freakish powers, which Drake later speculated was plagiarized by Stan Lee to create The X-Men. There was also the young Jim Shooter who purposely emulated Marvel's writing when he wrote for DC after much study of both companies' styles, such as for the Legion of Super-Heroes feature. In 1966, National Periodical Publications had set up its own television arm, led by Allen Ducovny to develop and produce projects for television, with Superman TV Corporation to handle its television distribution of NPP's TV shows.

A 1966 Batman TV show on the ABC network sparked a temporary spike in comic book sales, and a brief fad for superheroes in Saturday morning animation (Filmation created most of DC's initial cartoons) and other media. DC significantly lightened the tone of many DC comics—particularly Batman and Detective Comics—to better complement the "camp" tone of the TV series. This tone coincided with the famous "Go-Go Checks" checkerboard cover-dress which featured a black-and-white checkerboard strip (all DC books cover dated February 1966 until August 1967) at the top of each comic, a misguided attempt by then-managing editor Irwin Donenfeld to make DC's output "stand out on the newsracks". In particular, DC artist Carmine Infantino complained that the visual cover distinctiveness made DC's titles easier for readers to see and then avoid in favor of Marvel's titles.

In 1967, Batman artist Infantino (who had designed popular Silver Age characters Batgirl and the Phantom Stranger) rose from art director to become DC's editorial director. With the growing popularity of upstart rival Marvel Comics threatening to topple DC from its longtime number-one position in the comics industry, he attempted to infuse the company with more focus towards marketing new and existing titles and characters with more adult sensibilities towards an emerging older age group of superhero comic book fans that grew out of Marvel's efforts to market their superhero line to college-aged adults. He also recruited major talents such as ex-Marvel artist and Spider-Man co-creator Steve Ditko and promising newcomers Neal Adams and Denny O'Neil and replaced some existing DC editors with artist-editors, including Joe Kubert and Dick Giordano, to give DC's output a more artistic critical eye.

Kinney National / Warner Communications (1967–1990)
In 1967, National Periodical Publications was purchased by Kinney National Company, which purchased Warner Bros.-Seven Arts in 1969. Kinney National spun off its non-entertainment assets in 1972 (as National Kinney Corporation) and changed its name to Warner Communications Inc.

In 1970, Jack Kirby moved from Marvel Comics to DC, at the end of the Silver Age of Comics, in which Kirby's contributions to Marvel played a large, integral role.

As artist Gil Kane described:

Jack was the single most influential figure in the turnaround in Marvel's fortunes from the time he rejoined the company ... It wasn't merely that Jack conceived most of the characters that are being done, but ... Jack's point of view and philosophy of drawing became the governing philosophy of the entire publishing company and, beyond the publishing company, of the entire field ... [Marvel took] Jack and use[d] him as a primer. They would get artists ... and they taught them the ABCs, which amounted to learning Jack Kirby ... Jack was like the Holy Scripture and they simply had to follow him without deviation. That's what was told to me ... It was how they taught everyone to reconcile all those opposing attitudes to one single master point of view.

Given carte blanche to write and illustrate his own stories, he created a handful of thematically-linked series he called collectively "The Fourth World". In the existing series Superman's Pal Jimmy Olsen and in his own, newly-launched series New Gods, Mister Miracle, and The Forever People, Kirby introduced such enduring characters and concepts as arch-villain Darkseid and the other-dimensional realm Apokolips. Furthermore, Kirby intended their stories to be reprinted in collected editions, in a publishing format that was later called the trade paperback, which became a standard industry practice decades later. While sales were respectable, they did not meet DC management's initially high expectations, and also suffered from a lack of comprehension and internal support from Infantino. By 1973 the "Fourth World" was all cancelled, although Kirby's conceptions soon became integral to the broadening of the DC Universe, especially after the major toy-company, Kenner Products, judged them ideal for their action-figure adaptation of the DC Universe, the Super Powers Collection. Obligated by his contract, Kirby created other unrelated series for DC, including Kamandi, The Demon, and OMAC, before ultimately returning to Marvel Comics in 1976.

Bronze Age
Following the science-fiction innovations of the Silver Age, the comics of the 1970s and 1980s became known as the Bronze Age, as fantasy gave way to more naturalistic and sometimes darker themes. Illegal drug use, banned by the Comics Code Authority, explicitly appeared in comics for the first time in Marvel Comics' story "Green Goblin Reborn!" in The Amazing Spider-Man No. 96 (May 1971), and after the Code's updating in response, DC offered a drug-fueled storyline in writer Dennis O'Neil and artist Neal Adams' Green Lantern, beginning with the story "Snowbirds Don't Fly" in the retitled Green Lantern / Green Arrow No. 85 (September 1971), which depicted Speedy, the teen sidekick of superhero archer Green Arrow, as having become a heroin addict.

Jenette Kahn, a former children's magazine publisher, replaced Infantino as editorial director in January 1976. As it happened, her first task even before being formally hired, was to convince Bill Sarnoff, the head of Warner Publishing, to keep DC as a publishing concern, as opposed to simply managing their licensing of their properties. With that established, DC had attempted to compete with the now-surging Marvel by dramatically increasing its output and attempting to win the market by flooding it. This included launching series featuring such new characters as Firestorm and Shade, the Changing Man, as well as an increasing array of non-superhero titles, in an attempt to recapture the pre-Wertham days of post-War comicdom.

DC Comics
In 1977, the company officially changed its name to DC Comics. It had used the brand "Superman-DC" since the 1950s, and was colloquially known as DC Comics for years.

In June 1978, five months before the release of the first Superman movie, Kahn expanded the line further, increasing the number of titles and story pages, and raising the price from 35 cents to 50 cents. Most series received eight-page back-up features while some had full-length twenty-five-page stories. This was a move the company called the "DC Explosion". The move was not successful, however, and corporate parent Warner dramatically cut back on these largely unsuccessful titles, firing many staffers in what industry watchers dubbed "the DC Implosion". In September 1978, the line was dramatically reduced and standard-size books returned to 17-page stories but for a still increased 40 cents. By 1980, the books returned to 50 cents with a 25-page story count but the story pages replaced house ads in the books.

Seeking new ways to boost market share, the new team of publisher Kahn, vice president Paul Levitz, and managing editor Giordano addressed the issue of talent instability. To that end—and following the example of Atlas/Seaboard Comics and such independent companies as Eclipse Comics—DC began to offer royalties in place of the industry-standard work-for-hire agreement in which creators worked for a flat fee and signed away all rights, giving talent a financial incentive tied to the success of their work. As it happened, the implementation of these incentives proved opportune considering Marvel Comics' Editor-in-Chief, Jim Shooter, was alienating much of his company's creative staff with his authoritarian manner and major talents there went to DC like Roy Thomas, Gene Colan, Marv Wolfman, and George Perez.

In addition, emulating the era's new television form, the miniseries while addressing the matter of an excessive number of ongoing titles fizzling out within a few issues of their start, DC created the industry concept of the comic book limited series. This publishing format allowed for the deliberate creation of finite storylines within a more flexible publishing format that could showcase creations without forcing the talent into unsustainable open-ended commitments. The first such title was World of Krypton in 1979, and its positive results led to subsequent similar titles and later more ambitious productions like Camelot 3000 for the direct market in 1982.

These changes in policy shaped the future of the medium as a whole, and in the short term allowed DC to entice creators away from rival Marvel, and encourage stability on individual titles. In November 1980 DC launched the ongoing series The New Teen Titans, by writer Marv Wolfman and artist George Pérez, two popular talents with a history of success. Their superhero-team comic, superficially similar to Marvel's ensemble series X-Men, but rooted in DC history, earned significant sales in part due to the stability of the creative team, who both continued with the title for six full years. In addition, Wolfman and Pérez took advantage of the limited-series option to create a spin-off title, Tales of the New Teen Titans, to present origin stories of their original characters without having to break the narrative flow of the main series or oblige them to double their work load with another ongoing title.

Modern Age
This successful revitalization of the Silver Age Teen Titans led DC's editors to seek the same for the wider DC Universe. The result, the Wolfman/Pérez 12-issue limited series Crisis on Infinite Earths, gave the company an opportunity to realign and jettison some of the characters' complicated backstory and continuity discrepancies. A companion publication, two volumes entitled The History of the DC Universe, set out the revised history of the major DC characters. Crisis featured many key deaths that shaped the DC Universe for the following decades, and it separated the timeline of DC publications into pre- and post-"Crisis".

Meanwhile, a parallel update had started in the non-superhero and horror titles. Since early 1984, the work of British writer Alan Moore had revitalized the horror series The Saga of the Swamp Thing, and soon numerous British writers, including Neil Gaiman and Grant Morrison, began freelancing for the company. The resulting influx of sophisticated horror-fantasy material led to DC in 1993 establishing the Vertigo mature-readers imprint, which did not subscribe to the Comics Code Authority.

Two DC limited series, Batman: The Dark Knight Returns by Frank Miller and Watchmen by Moore and artist Dave Gibbons, drew attention in the mainstream press for their dark psychological complexity and promotion of the antihero. These titles helped pave the way for comics to be more widely accepted in literary-criticism circles and to make inroads into the book industry, with collected editions of these series as commercially successful trade paperbacks.

The mid-1980s also saw the end of many long-running DC war comics, including series that had been in print since the 1960s. These titles, all with over 100 issues, included Sgt. Rock, G.I. Combat, The Unknown Soldier, and Weird War Tales.

Time Warner / Time Warner Entertainment / AOL Time Warner (1990–2018)
In March 1989, Warner Communications merged with Time Inc., making DC Comics a subsidiary of Time Warner. In June, the first Tim Burton-directed Batman movie was released, and DC began publishing its hardcover series of DC Archive Editions, collections of many of their early, key comics series, featuring rare and expensive stories unseen by many modern fans. Restoration for many of the Archive Editions was handled by Rick Keene with colour restoration by DC's long-time resident colourist, Bob LeRose. These collections attempted to retroactively credit many of the writers and artists who had worked without much recognition for DC during the early period of comics when individual credits were few and far between.

The comics industry experienced a brief boom in the early 1990s, thanks to a combination of speculative purchasing (mass purchase of the books as collectible items, with intent to resell at a higher value as the rising value of older issues, was thought to imply that all comics would rise dramatically in price) and several storylines which gained attention from the mainstream media. DC's extended storylines in which Superman was killed, Batman was crippled and superhero Green Lantern turned into the supervillain Parallax resulted in dramatically increased sales, but the increases were as temporary as the hero's replacements. Sales dropped off as the industry went into a major slump, while manufactured "collectables" numbering in the millions replaced quality with quantity until fans and speculators alike deserted the medium in droves.

DC's Piranha Press and other imprints (including the mature readers line Vertigo, and Helix, a short-lived science fiction imprint) were introduced to facilitate compartmentalized diversification and allow for specialized marketing of individual product lines. They increased the use of non-traditional contractual arrangements, including the dramatic rise of creator-owned projects, leading to a significant increase in critically lauded work (much of it for Vertigo) and the licensing of material from other companies. DC also increased publication of book-store friendly formats, including trade paperback collections of individual serial comics, as well as original graphic novels.

One of the other imprints was Impact Comics from 1991 to 1992 in which the Archie Comics superheroes were licensed and revamped. The stories in the line were part of its own shared universe.

DC entered into a publishing agreement with Milestone Media that gave DC a line of comics featuring a culturally and racially diverse range of superhero characters. Although the Milestone line ceased publication after a few years, it yielded the popular animated series Static Shock. DC established Paradox Press to publish material such as the large-format Big Book of... series of multi-artist interpretations on individual themes, and such crime fiction as the graphic novel Road to Perdition. In 1998, DC purchased WildStorm Comics, Jim Lee's imprint under the Image Comics banner, continuing it for many years as a wholly separate imprint – and fictional universe – with its own style and audience. As part of this purchase, DC also began to publish titles under the fledgling WildStorm sub-imprint America's Best Comics (ABC), a series of titles created by Alan Moore, including The League of Extraordinary Gentlemen, Tom Strong, and Promethea. Moore strongly contested this situation, and DC eventually stopped publishing ABC.

In March 2003 DC acquired publishing and merchandising rights to the long-running fantasy series Elfquest, previously self-published by creators Wendy and Richard Pini under their WaRP Graphics publication banner. This series then followed another non-DC title, Tower Comics' series T.H.U.N.D.E.R. Agents, in collection into DC Archive Editions. In 2004 DC temporarily acquired the North American publishing rights to graphic novels from European publishers 2000 AD and Humanoids. It also rebranded its younger-audience titles with the mascot Johnny DC and established the CMX imprint to reprint translated manga. In 2006, CMX took over from Dark Horse Comics publication of the webcomic Megatokyo in print form. DC also took advantage of the demise of Kitchen Sink Press and acquired the rights to much of the work of Will Eisner, such as his The Spirit series and his graphic novels.

In 2004, DC began laying the groundwork for a full continuity-reshuffling sequel to Crisis on Infinite Earths, promising substantial changes to the DC Universe (and side-stepping the 1994 Zero Hour event which similarly tried to ret-con the history of the DCU). In 2005, the critically lauded Batman Begins film was released; also, the company published several limited series establishing increasingly escalated conflicts among DC's heroes, with events climaxing in the Infinite Crisis limited series. Immediately after this event, DC's ongoing series jumped forward a full year in their in-story continuity, as DC launched a weekly series, 52, to gradually fill in the missing time. Concurrently, DC lost the copyright to "Superboy" (while retaining the trademark) when the heirs of Jerry Siegel used a provision of the 1976 revision to the copyright law to regain ownership.

In 2005, DC launched its "All-Star" line (evoking the title of the 1940s publication), designed to feature some of the company's best-known characters in stories that eschewed the long and convoluted continuity of the DC Universe. The line began with All-Star Batman & Robin the Boy Wonder and All-Star Superman, with All-Star Wonder Woman and All-Star Batgirl announced in 2006 but neither being released nor scheduled as of the end of 2009.

DC licensed characters from the Archie Comics imprint Red Circle Comics by 2007. They appeared in the Red Circle line, based in the DC Universe, with a series of one-shots followed by a miniseries that lead into two ongoing titles, each lasting 10 issues.

DC Entertainment
In 2011, DC rebooted all of its running titles following the Flashpoint storyline. The reboot called The New 52 gave new origin stories and costume designs to many of DC's characters.

DC licensed pulp characters including Doc Savage and the Spirit which it then used, along with some DC heroes, as part of the First Wave comics line launched in 2010 and lasting through fall 2011.

In May 2011, DC announced it would begin releasing digital versions of their comics on the same day as paper versions.

On June 1, 2011, DC announced that it would end all ongoing series set in the DC Universe in August and relaunch its comic line with 52 issue #1s, starting with Justice League on August 31 (written by Geoff Johns and drawn by Jim Lee), with the rest to follow later on in September.

On June 4, 2013, DC unveiled two new digital comic innovations to enhance interactivity: DC2 and DC2 Multiverse. DC2 layers dynamic artwork onto digital comic panels, adding a new level of dimension to digital storytelling, while DC2 Multiverse allows readers to determine a specific story outcome by selecting individual characters, storylines and plot developments while reading the comic, meaning one digital comic has multiple outcomes. DC2 appeared in the digital-first title, Batman '66, based on the 1960s television series and DC2 Multiverse appeared in Batman: Arkham Origins, a digital-first title based on the video game of the same name.

In 2014, DC announced an eight-issue miniseries titled Convergence which began in April 2015.

In 2016, DC announced a line-wide relaunch titled DC Rebirth. The new line would launch with an 80-page one-shot titled DC Universe: Rebirth, written by Geoff Johns, with art from Gary Frank, Ethan Van Sciver, and more. After that, many new series would launch with a twice-monthly release schedule and new creative teams for nearly every title. The relaunch was meant to bring back the legacy and heart many felt had been missing from DC characters since the launch of the New 52. Rebirth brought huge success, both financially and critically.

WarnerMedia / Warner Bros. Discovery unit (2018–present)
On February 21, 2020, the Co-Publisher of DC Comics, Dan DiDio stepped down after 10 years at that position. The company did not give a reason for the move, nor did it indicate whether it was his decision or the company's. The leadership change was the latest event in the company restructuring which began the previous month, as several top executives were laid off from the company. However, Bleeding Cool reported that he was fired.

In June 2020, Warner Bros. announced a separate DC-themed online-only convention. Known as DC FanDome, the free "immersive virtual fan experience" was a 24-hour-long event held on August 22, 2020. The main presentation, entitled "DC FanDome: Hall of Heroes", was held as scheduled on August 22. The remaining programming was provided through a one-day video on demand experience, "DC FanDome: Explore the Multiverse", on September 12.

As Warner Bros. and DC's response to San Diego Comic-Con's cancellation due to the COVID-19 pandemic, the convention featured information about DC-based content including the DC Extended Universe film franchise, the Arrowverse television franchise, comic books, and video games. The convention also returned for the virtual premiere of Wonder Woman 1984 and returned once again on October 16, 2021.

In August 2020, roughly one-third of DC's editorial ranks were laid off, including the editor-in-chief, senior story editor, executive editor, and several senior VPs.

In March 2021, DC relaunched their entire line once again under the banner of Infinite Frontier. After the events of the Dark Nights: Death Metal storyline, the DC Multiverse was expanded into a larger "Omniverse" where everything is canon, effectively reversing the changes The New 52 introduced a decade prior.

Furthermore, AT&T spun off WarnerMedia to Discovery, forming Warner Bros. Discovery. This merger was completed on April 8, 2022.

In January 2023, DC relaunched their line under the banner of Dawn of DC following the conclusion of Dark Crisis on Infinite Earths and Lazarus Planet. Later that year, Jim Lee was promoted to President of DC in May.

Logo

 * DC's first logo appeared on the April 1940 issues of its titles. The small logo, with no background, read simply, "A DC Publication".
 * The November 1941 DC titles introduced an updated logo. This version was almost twice the size of the previous one and was the first version with a white background. The name "Superman" was added to "A DC Publication", effectively acknowledging both Superman and Batman. This logo was the first to occupy the top-left corner of the cover, where the logo has usually resided since. The company now referred to itself in its advertising as "Superman-DC".
 * In November 1949, the logo was modified to incorporate the company's formal name, National Comics Publications. This logo also served as the round body of Johnny DC, DC's mascot in the 1960s.
 * In October 1970, DC briefly retired the circular logo in favour of a simple "DC" in a rectangle with the name of the title, or the star of the book; the logo on many issues of Action Comics, for example, read "DC Superman". An image of the lead character either appeared above or below the rectangle. For books that did not have a single star, such as anthologies like House of Mystery or team series such as Justice League of America, the title and "DC" appeared in a stylized logo, such as a bat for "House of Mystery". This use of characters as logos helped to establish the likenesses as trademarks, and was similar to Marvel's contemporaneous use of characters as part of its cover branding.
 * DC's "100 Page Super-Spectacular" titles and later 100-page and "Giant" issues published from 1972 to 1974 featured a logo exclusive to these editions: the letters "DC" in a simple sans-serif typeface within a circle. A variant had the letters in a square.
 * The July 1972 DC titles featured a new circular logo. The letters "DC" were rendered in a block-like typeface that remained through later logo revisions until 2005. The title of the book usually appeared inside the circle, either above or below the letters.
 * In December 1973, this logo was modified with the addition of the words "The Line of DC Super-Stars" and the star motif that continued in later logos. This logo was placed in the top center of the cover from August 1975 to October 1976.
 * When Jenette Kahn became DC's publisher in late 1976, she commissioned graphic designer Milton Glaser to design a new logo. Popularly referred to as the "DC bullet", this logo premiered on the February 1977 titles. Although it varied in size and colour and was at times cropped by the edges of the cover, or briefly rotated 4 degrees, it remained essentially unchanged for nearly three decades. Despite logo changes since 2005, the old "DC bullet" continues to be used only on the DC Archive Editions series.
 * On May 8, 2005, a new logo (dubbed the "DC spin") was unveiled, debuting on DC titles in June 2005 with DC Special: The Return of Donna Troy No. 1 and the rest of the titles the following week. In addition to comics, it was designed for DC properties in other media, which was used for movies since Batman Begins, with Superman Returns showing the logo's normal variant, and the TV series Smallville, the animated series Justice League Unlimited and others, as well as for collectibles and other merchandise. The logo was designed by Josh Beatman of Brainchild Studios and DC executive Richard Bruning.
 * In March 2012, DC unveiled a new logo consisting of the letter "D" flipping back to reveal the letter "C" and "DC ENTERTAINMENT".
 * DC Entertainment announced a new identity and logo for another iconic DC Comics universe brand on May 17, 2016. The new logo was first used on May 25, 2016, in conjunction with the release of DC Universe: Rebirth Special #1 by Geoff Johns.

Active

 * DC (1937–present)
 * Elseworlds (1989–2010, 2024–present)
 * WildStorm (1999–2010, 2017–present)
 * Earth-M (1993–1997, 2018–present)
 * Young Animal (2016–present)
 * DC Black Label (2018–present)
 * Sandman Universe (2018–present)
 * Hill House Comics (2019–present)
 * Wonder Comics (2019–present)
 * DC Graphic Novels for Young Adults (2020–present)
 * DC Graphic Novels for Kids (2020–present)
 * DC Compact Comics (2024–present)
 * Mad (1953–present)

Defunct

 * DC Archive Editions (1989–2014; replaced by DC Omnibus)
 * Piranha Press (1989–1993; renamed Paradox Press)
 * Impact Comics (1991–1993; licensed from Archie Comics)
 * Vertigo Comics (1993–2019)
 * Amalgam Comics (1996–1997; jointly with Marvel Comics)
 * Helix (1996–1998; merged with Vertigo)
 * Tangent Comics (1997–1998)
 * Paradox Press (1998–2003)
 * WildStorm Productions (1999–2010)
 * America's Best Comics (1999–2005)
 * Homage Comics (1999–2004; merged to form WildStorm Signature)
 * Cliffhanger (1999–2004; merged to form WildStorm Signature)
 * WildStorm Signature (2004–2006; merged with main WildStorm line)
 * CMX Manga (2004–2010)
 * DC Focus (2004–2005; merged with main DC line)
 * Johnny DC (2004–2012)
 * All Star (2005–2008)
 * Minx (2007–2008)
 * Zuda Comics (2007–2010)
 * First Wave (2010–2011; licensed from Condé Nast Publications and Will Eisner Library)
 * DC Ink (2019; replaced by DC Graphic Novels for Young Adults)
 * DC Zoom (2019; replaced by DC Graphic Novels for Kids)

title: DC Comics


text:
Diophantine equation

In mathematics, a Diophantine equation is an equation, typically a polynomial equation in two or more unknowns with integer coefficients, for which only integer solutions are of interest. A linear Diophantine equation equates to a constant the sum of two or more monomials, each of degree one. An exponential Diophantine equation is one in which unknowns can appear in exponents.

Diophantine problems have fewer equations than unknowns and involve finding integers that solve simultaneously all equations. As such systems of equations define algebraic curves, algebraic surfaces, or, more generally, algebraic sets, their study is a part of algebraic geometry that is called Diophantine geometry.

The word Diophantine refers to the Hellenistic mathematician of the 3rd century, Diophantus of Alexandria, who made a study of such equations and was one of the first mathematicians to introduce symbolism into algebra. The mathematical study of Diophantine problems that Diophantus initiated is now called Diophantine analysis.

While individual equations present a kind of puzzle and have been considered throughout history, the formulation of general theories of Diophantine equations (beyond the case of linear and quadratic equations) was an achievement of the twentieth century.

Examples
In the following Diophantine equations, $w, x, y$, and $z$ are the unknowns and the other letters are given constants:

One equation
The simplest linear Diophantine equation takes the form $$ax+by=c,$$ where $n$, $n$ and $x, y$ are given integers. The solutions are described by the following theorem:
 * This Diophantine equation has a solution (where $z$ and $a$ are integers) if and only if $b$ is a multiple of the greatest common divisor of $c$ and $x$. Moreover, if $12^{3} + 1^{3} = 9^{3} + 10^{3} = 1729$ is a solution, then the other solutions have the form $n = 2$, where $y$ is an arbitrary integer, and $c$ and $a$ are the quotients of $b$ and $k$ (respectively) by the greatest common divisor of $u$ and $v$.

Proof: If $a$ is this greatest common divisor, Bézout's identity asserts the existence of integers $b$ and $a$ such that $(x, y, z)$. If $b$ is a multiple of $d$, then $(x, y, z)$ for some integer $e$, and $95800^{4} + 217519^{4} + 414560^{4} = 422481^{4}$ is a solution. On the other hand, for every pair of integers $f$ and $c$, the greatest common divisor $d$ of $h$ and $x$ divides $(x, y)$. Thus, if the equation has a solution, then $y$ must be a multiple of $d$. If $(x + kv, y − ku)$ and $ae + bf = d$, then for every solution $c = dh$, we have $$\begin{align} a(x+kv) + b(y-ku) &= ax+by+k(av-bu) \\ &= ax+by+k(udv-vdu) \\ &= ax+by, \end{align}$$ showing that $(eh, fh)$ is another solution. Finally, given two solutions such that $$ax_1 + by_1 = ax_2 + by_2 = c,$$ one deduces that $$u(x_2 - x_1) + v(y_2 - y_1) = 0.$$ As $a$ and $b$ are coprime, Euclid's lemma shows that $c$ divides $ax + by$, and thus that there exists an integer $d$ such that both $$x_2 - x_1 = kv, \quad y_2 - y_1 = -ku.$$ Therefore, $$x_2 = x_1 + kv, \quad y_2 = y_1 - ku,$$ which completes the proof.

Chinese remainder theorem
The Chinese remainder theorem describes an important class of linear Diophantine systems of equations: let $$n_1, \dots, n_k$$ be $u$ pairwise coprime integers greater than one, $$a_1, \dots, a_k$$ be $v$ arbitrary integers, and $v$ be the product $$n_1 \cdots n_k.$$ The Chinese remainder theorem asserts that the following linear Diophantine system has exactly one solution $$(x, x_1, \dots, x_k)$$ such that $a = ud$, and that the other solutions are obtained by adding to $k$ a multiple of $k$: $$\begin{align} x &= a_1 + n_1\,x_1\\ &\;\;\vdots\\ x &= a_k + n_k\,x_k \end{align}$$

System of linear Diophantine equations
More generally, every system of linear Diophantine equations may be solved by computing the Smith normal form of its matrix, in a way that is similar to the use of the reduced row echelon form to solve a system of linear equations over a field. Using matrix notation every system of linear Diophantine equations may be written $$AX = C,$$ where $k$ is an $b = vd$ matrix of integers, $N$ is an $(x, y)$ column matrix of unknowns and $x$ is an $(x + kv, y − ku)$ column matrix of integers.

The computation of the Smith normal form of $N$ provides two unimodular matrices (that is matrices that are invertible over the integers and have ±1 as determinant) $A$ and $X$ of respective dimensions $x_{2} − x_{1}$ and $0 ≤ x < N$, such that the matrix $$B = [b_{i,j}] = UAV$$ is such that $C$ is not zero for $A$ not greater than some integer $U$, and all the other entries are zero. The system to be solved may thus be rewritten as $$B (V^{-1}X) = UC.$$ Calling $V$ the entries of $m × n$ and $b_{i,i}$ those of $n × 1$, this leads to the system $$\begin{align} & b_{i,i}y_i = d_i, \quad 1 \leq i \leq k \\ & 0y_i = d_i, \quad k < i \leq n. \end{align}$$

This system is equivalent to the given one in the following sense: A column matrix of integers $i$ is a solution of the given system if and only if $m × 1$ for some column matrix of integers $k$ such that $m × m$.

It follows that the system has a solution if and only if $y_{i}$ divides $d_{i}$ for $n × n$ and $VX$ for $D = UC$. If this condition is fulfilled, the solutions of the given system are $$ V\, \begin{bmatrix} \frac{d_1}{b_{1,1}}\\ \vdots\\ \frac{d_k}{b_{k,k}}\\ h_{k+1}\\ \vdots\\ h_n \end{bmatrix}\,, $$ where $x = Vy$ are arbitrary integers.

Hermite normal form may also be used for solving systems of linear Diophantine equations. However, Hermite normal form does not directly provide the solutions; to get the solutions from the Hermite normal form, one has to successively solve several linear equations. Nevertheless, Richard Zippel wrote that the Smith normal form "is somewhat more than is actually needed to solve linear diophantine equations. Instead of reducing the equation to diagonal form, we only need to make it triangular, which is called the Hermite normal form. The Hermite normal form is substantially easier to compute than the Smith normal form."

Integer linear programming amounts to finding some integer solutions (optimal in some sense) of linear systems that include also inequations. Thus systems of linear Diophantine equations are basic in this context, and textbooks on integer programming usually have a treatment of systems of linear Diophantine equations.

Homogeneous equations
A homogeneous Diophantine equation is a Diophantine equation that is defined by a homogeneous polynomial. A typical such equation is the equation of Fermat's Last Theorem
 * $$x^d+y^d -z^d=0.$$

As a homogeneous polynomial in $x$ indeterminates defines a hypersurface in the projective space of dimension $By = D$, solving a homogeneous Diophantine equation is the same as finding the rational points of a projective hypersurface.

Solving a homogeneous Diophantine equation is generally a very difficult problem, even in the simplest non-trivial case of three indeterminates (in the case of two indeterminates the problem is equivalent with testing if a rational number is the $y$th power of another rational number). A witness of the difficulty of the problem is Fermat's Last Theorem (for $i ≤ k$, there is no integer solution of the above equation), which needed more than three centuries of mathematicians' efforts before being solved.

For degrees higher than three, most known results are theorems asserting that there are no solutions (for example Fermat's Last Theorem) or that the number of solutions is finite (for example Falting's theorem).

For the degree three, there are general solving methods, which work on almost all equations that are encountered in practice, but no algorithm is known that works for every cubic equation.

Degree two
Homogeneous Diophantine equations of degree two are easier to solve. The standard solving method proceeds in two steps. One has first to find one solution, or to prove that there is no solution. When a solution has been found, all solutions are then deduced.

For proving that there is no solution, one may reduce the equation modulo $b_{i,i}$. For example, the Diophantine equation
 * $$x^2+y^2=3z^2,$$

does not have any other solution than the trivial solution $d_{i} = 0$. In fact, by dividing $d_{i}$, and $n$ by their greatest common divisor, one may suppose that they are coprime. The squares modulo 4 are congruent to 0 and 1. Thus the left-hand side of the equation is congruent to 0, 1, or 2, and the right-hand side is congruent to 0 or 3. Thus the equality may be obtained only if $d$, and $p$ are all even, and are thus not coprime. Thus the only solution is the trivial solution $i > k$. This shows that there is no rational point on a circle of radius $$\sqrt{3},$$ centered at the origin.

More generally, the Hasse principle allows deciding whether a homogeneous Diophantine equation of degree two has an integer solution, and computing a solution if there exist.

If a non-trivial integer solution is known, one may produce all other solutions in the following way.

Geometric interpretation
Let
 * $$Q(x_1, \ldots, x_n)=0$$

be a homogeneous Diophantine equation, where $$Q(x_1, \ldots, x_n)$$ is a quadratic form (that is, a homogeneous polynomial of degree 2), with integer coefficients. The trivial solution is the solution where all $$x_i$$ are zero. If $$(a_1, \ldots, a_n)$$ is a non-trivial integer solution of this equation, then $$\left(a_1, \ldots, a_n\right)$$ are the homogeneous coordinates of a rational point of the hypersurface defined by $x, y$. Conversely, if $\left(\frac {p_1}q, \ldots, \frac {p_n}q \right)$  are homogeneous coordinates of a rational point of this hypersurface, where $$q, p_1, \ldots, p_n$$ are integers, then $$\left(p_1, \ldots, p_n\right)$$ is an integer solution of the Diophantine equation. Moreover, the integer solutions that define a given rational point are all sequences of the form
 * $$\left(k\frac{p_1}d, \ldots, k\frac{p_n}d\right),$$

where $z$ is any integer, and $x, y$ is the greatest common divisor of the $$p_i.$$

It follows that solving the Diophantine equation $$Q(x_1, \ldots, x_n)=0$$ is completely reduced to finding the rational points of the corresponding projective hypersurface.

Parameterization
Let now $$A=\left(a_1, \ldots, a_n\right)$$ be an integer solution of the equation $$Q(x_1, \ldots, x_n)=0.$$ As $z$ is a polynomial of degree two, a line passing through $Q$ crosses the hypersurface at a single other point, which is rational if and only if the line is rational (that is, if the line is defined by rational parameters). This allows parameterizing the hypersurface by the lines passing through $k$, and the rational points are those that are obtained from rational lines, that is, those that correspond to rational values of the parameters.

More precisely, one may proceed as follows.

By permuting the indices, one may suppose, without loss of generality that $$a_n\ne 0.$$ Then one may pass to the affine case by considering the affine hypersurface defined by
 * $$q(x_1,\ldots,x_{n-1})=Q(x_1, \ldots, x_{n-1},1),$$

which has the rational point
 * $$R= (r_1, \ldots, r_{n-1})=\left(\frac{a_1}{a_n}, \ldots, \frac{a_{n-1}}{a_n}\right).$$

If this rational point is a singular point, that is if all partial derivatives are zero at $d$, all lines passing through $Q$ are contained in the hypersurface, and one has a cone. The change of variables
 * $$y_i=x_i-r_i$$

does not change the rational points, and transforms $A$ into a homogeneous polynomial in $h_{k+1}, …, h_{n}$ variables. In this case, the problem may thus be solved by applying the method to an equation with fewer variables.

If the polynomial $A$ is a product of linear polynomials (possibly with non-rational coefficients), then it defines two hyperplanes. The intersection of these hyperplanes is a rational flat, and contains rational singular points. This case is thus a special instance of the preceding case.

In the general case, consider the parametric equation of a line passing through $R$:
 * $$\begin{align}

x_2 &= r_2 + t_2(x_1-r_1)\\ &\;\;\vdots\\ x_{n-1} &= r_{n-1} + t_{n-1}(x_1-r_1). \end{align}$$ Substituting this in $R$, one gets a polynomial of degree two in $n − 1$, that is zero for $d > 2$. It is thus divisible by $(0, 0, 0)$. The quotient is linear in $(0, 0, 0)$, and may be solved for expressing $n − 1$ as a quotient of two polynomials of degree at most two in $$t_2, \ldots, t_{n-1},$$ with integer coefficients:
 * $$x_1=\frac{f_1(t_2, \ldots, t_{n-1})}{f_n(t_2, \ldots, t_{n-1})}.$$

Substituting this in the expressions for $$x_2, \ldots, x_{n-1},$$ one gets, for $x1$,
 * $$x_i=\frac{f_i(t_2, \ldots, t_{n-1})}{f_n(t_2, \ldots, t_{n-1})},$$

where $$f_1, \ldots, f_n$$ are polynomials of degree at most two with integer coefficients.

Then, one can return to the homogeneous case. Let, for $x1 = r1$,
 * $$F_i(t_1, \ldots, t_{n-1})=t_1^2 f_i\left(\frac{t_2}{t_1}, \ldots, \frac{t_{n-1}}{t_1} \right),$$

be the homogenization of $$f_i.$$ These quadratic polynomials with integer coefficients form a parameterization of the projective hypersurface defined by $q$:
 * $$\begin{align}

x_1&= F_1(t_1, \ldots, t_{n-1})\\ &\;\;\vdots\\ x_n&= F_n(t_1, \ldots, t_{n-1}). \end{align}$$

A point of the projective hypersurface defined by $q$ is rational if and only if it may be obtained from rational values of $$t_1, \ldots, t_{n-1}.$$ As $$F_1, \ldots,F_n$$ are homogeneous polynomials, the point is not changed if all $R$ are multiplied by the same rational number. Thus, one may suppose that $$t_1, \ldots, t_{n-1}$$ are coprime integers. It follows that the integer solutions of the Diophantine equation are exactly the sequences $$(x_1, \ldots, x_n)$$ where, for $x1 – r1$,
 * $$x_i= k\,\frac{F_i(t_1, \ldots, t_{n-1})}{d},$$

where $q$ is an integer, $$t_1, \ldots, t_{n-1}$$ are coprime integers, and $Q$ is the greatest common divisor of the $Q$ integers $$F_i(t_1, \ldots, t_{n-1}).$$

One could hope that the coprimality of the $ti$, could imply that $x1$. Unfortunately this is not the case, as shown in the next section.

Example of Pythagorean triples
The equation
 * $$x^2+y^2-z^2=0$$

is probably the first homogeneous Diophantine equation of degree two that has been studied. Its solutions are the Pythagorean triples. This is also the homogeneous equation of the unit circle. In this section, we show how the above method allows retrieving Euclid's formula for generating Pythagorean triples.

For retrieving exactly Euclid's formula, we start from the solution $x1$, corresponding to the point $i = 1, …, n − 1$ of the unit circle. A line passing through this point may be parameterized by its slope:
 * $$y=t(x+1).$$

Putting this in the circle equation
 * $$x^2+y^2-1=0,$$

one gets
 * $$x^2-1 +t^2(x+1)^2=0.$$

Dividing by $i = 1, …, n$, results in
 * $$x-1+t^2(x+1)=0,$$

which is easy to solve in $k$:
 * $$x=\frac{1-t^2}{1+t^2}.$$

It follows
 * $$y=t(x+1) = \frac{2t}{1+t^2}.$$

Homogenizing as described above one gets all solutions as
 * $$\begin{align}

x&=k\,\frac{s^2-t^2}{d}\\ y&=k\,\frac{2st}{d}\\ z&=k\,\frac{s^2+t^2}{d}, \end{align}$$ where $d$ is any integer, $n$ and $ti$ are coprime integers, and $x$ is the greatest common divisor of the three numerators. In fact, $i = 1, ..., n$ if $k$ and $s$ are both odd, and $d = 1$ if one is odd and the other is even.

The primitive triples are the solutions where $(−1, 0, 1)$ and $(−1, 0)$.

This description of the solutions differs slightly from Euclid's formula because Euclid's formula considers only the solutions such that $t$, and $d$ are all positive, and does not distinguish between two triples that differ by the exchange of $s$ and $t$,

Typical questions
The questions asked in Diophantine analysis include:

 * 1) Are there any solutions?
 * 2) Are there any solutions beyond some that are easily found by inspection?
 * 3) Are there finitely or infinitely many solutions?
 * 4) Can all solutions be found in theory?
 * 5) Can one in practice compute a full list of solutions?

These traditional problems often lay unsolved for centuries, and mathematicians gradually came to understand their depth (in some cases), rather than treat them as puzzles.

Typical problem
The given information is that a father's age is 1 less than twice that of his son, and that the digits $x, y$ making up the father's age are reversed in the son's age (i.e. $z$). This leads to the equation $x + 1$, thus $d = 2$. Inspection gives the result $d = 1$, $k = 1$, and thus $x$ equals 73 years and $y$ equals 37 years. One may easily show that there is not any other solution with $AB$ and $BA$ positive integers less than 10.

Many well known puzzles in the field of recreational mathematics lead to diophantine equations. Examples include the cannonball problem, Archimedes's cattle problem and the monkey and the coconuts.

17th and 18th centuries
In 1637, Pierre de Fermat scribbled on the margin of his copy of Arithmetica: "It is impossible to separate a cube into two cubes, or a fourth power into two fourth powers, or in general, any power higher than the second into two like powers." Stated in more modern language, "The equation $s > t > 0$ has no solutions for any $AB$ higher than 2." Following this, he wrote: "I have discovered a truly marvelous proof of this proposition, which this margin is too narrow to contain." Such a proof eluded mathematicians for centuries, however, and as such his statement became famous as Fermat's Last Theorem. It was not until 1995 that it was proven by the British mathematician Andrew Wiles.

In 1657, Fermat attempted to solve the Diophantine equation $10A + B = 2(10B + A) − 1$ (solved by Brahmagupta over 1000 years earlier). The equation was eventually solved by Euler in the early 18th century, who also solved a number of other Diophantine equations. The smallest solution of this equation in positive integers is $19B − 8A = 1$, $A = 7$ (see Chakravala method).

Hilbert's tenth problem
In 1900, David Hilbert proposed the solvability of all Diophantine equations as the tenth of his fundamental problems. In 1970, Yuri Matiyasevich solved it negatively, building on work of Julia Robinson, Martin Davis, and Hilary Putnam to prove that a general algorithm for solving all Diophantine equations cannot exist.

Diophantine geometry
Diophantine geometry, is the application of techniques from algebraic geometry which considers equations that also have a geometric meaning. The central idea of Diophantine geometry is that of a rational point, namely a solution to a polynomial equation or a system of polynomial equations, which is a vector in a prescribed field $BA$, when $A$ is not algebraically closed.

Modern research
The oldest general method for solving a Diophantine equation—or for proving that there is no solution— is the method of infinite descent, which was introduced by Pierre de Fermat. Another general method is the Hasse principle that uses modular arithmetic modulo all prime numbers for finding the solutions. Despite many improvements these methods cannot solve most Diophantine equations.

The difficulty of solving Diophantine equations is illustrated by Hilbert's tenth problem, which was set in 1900 by David Hilbert; it was to find an algorithm to determine whether a given polynomial Diophantine equation with integer coefficients has an integer solution. Matiyasevich's theorem implies that such an algorithm cannot exist.

During the 20th century, a new approach has been deeply explored, consisting of using algebraic geometry. In fact, a Diophantine equation can be viewed as the equation of an hypersurface, and the solutions of the equation are the points of the hypersurface that have integer coordinates.

This approach led eventually to the proof by Andrew Wiles in 1994 of Fermat's Last Theorem, stated without proof around 1637. This is another illustration of the difficulty of solving Diophantine equations.

Infinite Diophantine equations
An example of an infinite Diophantine equation is: $$n = a^2 + 2b^2 + 3c^2 + 4d^2 + 5e^2 + \cdots,$$ which can be expressed as "How many ways can a given integer $B$ be written as the sum of a square plus twice a square plus thrice a square and so on?" The number of ways this can be done for each $n$ forms an integer sequence. Infinite Diophantine equations are related to theta functions and infinite dimensional lattices. This equation always has a solution for any positive $K$. Compare this to: $$n = a^2 + 4b^2 + 9c^2 + 16d^2 + 25e^2 + \cdots,$$ which does not always have a solution for positive $K$.

Exponential Diophantine equations
If a Diophantine equation has as an additional variable or variables occurring as exponents, it is an exponential Diophantine equation. Examples include:

 * the Ramanujan–Nagell equation, $B = 3$
 * the equation of the Fermat–Catalan conjecture and Beal's conjecture, $an + bn = cn$ with inequality restrictions on the exponents
 * the Erdős–Moser equation, $61x^{2} + 1 = y^{2}$

A general theory for such equations is not available; particular cases such as Catalan's conjecture and Fermat's Last Theorem have been tackled. However, the majority are solved via ad-hoc methods such as Størmer's theorem or even trial and error.

title: Diophantine equation


text:
Diophantus

Diophantus of Alexandria (born c. AD 200; died c. AD 284) was a Greek mathematician, who was the author of two main works: On Polygonal Numbers, which survives incomplete, and the Arithmetica in thirteen books, most of it extant, made up of arithmetical problems that are solved through algebraic equations.

Diophantus was the first Greek mathematician who recognized positive rational numbers as numbers, by allowing fractions for coefficients and solutions. He coined the term παρισότης (parisotes) to refer to an approximate equality. This term was rendered as adaequalitas in Latin, and became the technique of adequality developed by Pierre de Fermat to find maxima for functions and tangent lines to curves.

Although not the earliest, the Arithmetica has the best-known use of algebraic notation to solve arithmetical problems coming from Greek antiquity, and some of its problems served as inspiration for later mathematicians working in analysis and number theory. In modern use, Diophantine equations are algebraic equations with integer coefficients for which integer solutions are sought. Diophantine geometry and Diophantine approximations are two other subareas of number theory that are named after him.

Biography
Diophantus was born into a Greek family and is known to have lived in Alexandria, Egypt, during the Roman era, between AD 200 and 214 to 284 or 298. Much of our knowledge of the life of Diophantus is derived from a 5th-century Greek anthology of number games and puzzles created by Metrodorus. One of the problems (sometimes called his epitaph) states:"Here lies Diophantus,' the wonder behold. Through art algebraic, the stone tells how old: 'God gave him his boyhood one-sixth of his life, One twelfth more as youth while whiskers grew rife; And then yet one-seventh ere marriage begun; In five years there came a bouncing new son. Alas, the dear child of master and sage After attaining half the measure of his father's life chill fate took him. After consoling his fate by the science of numbers for four years, he ended his life.'"This puzzle implies that Diophantus' age $x$ can be expressed as

which gives $x = x⁄6 + x⁄12 + x⁄7 + 5 + x⁄2 + 4$ a value of 84 years. However, the accuracy of the information cannot be confirmed.

In popular culture, this puzzle was the Puzzle No.142 in Professor Layton and Pandora's Box as one of the hardest solving puzzles in the game, which needed to be unlocked by solving other puzzles first.

Arithmetica

Arithmetica is the major work of Diophantus and the most prominent work on premodern algebra in Greek mathematics. It is a collection of problems giving numerical solutions of both determinate and indeterminate equations. Of the original thirteen books of which Arithmetica consisted only six have survived, though there are some who believe that four Arabic books discovered in 1968 are also by Diophantus. Some Diophantine problems from Arithmetica have been found in Arabic sources.

It should be mentioned here that Diophantus never used general methods in his solutions. Hermann Hankel, renowned German mathematician made the following remark regarding Diophantus:"Our author (Diophantos) not the slightest trace of a general, comprehensive method is discernible; each problem calls for some special method which refuses to work even for the most closely related problems. For this reason it is difficult for the modern scholar to solve the 101st problem even after having studied 100 of Diophantos's solutions."

History
Like many other Greek mathematical treatises, Diophantus was forgotten in Western Europe during the Dark Ages, since the study of ancient Greek, and literacy in general, had greatly declined. The portion of the Greek Arithmetica that survived, however, was, like all ancient Greek texts transmitted to the early modern world, copied by, and thus known to, medieval Byzantine scholars. Scholia on Diophantus by the Byzantine Greek scholar John Chortasmenos (1370–1437) are preserved together with a comprehensive commentary written by the earlier Greek scholar Maximos Planudes (1260 – 1305), who produced an edition of Diophantus within the library of the Chora Monastery in Byzantine Constantinople. In addition, some portion of the Arithmetica probably survived in the Arab tradition (see above). In 1463 German mathematician Regiomontanus wrote:"No one has yet translated from the Greek into Latin the thirteen books of Diophantus, in which the very flower of the whole of arithmetic lies hidden."Arithmetica was first translated from Greek into Latin by Bombelli in 1570, but the translation was never published. However, Bombelli borrowed many of the problems for his own book Algebra. The editio princeps of Arithmetica was published in 1575 by Xylander. The Latin translation of Arithmetica by Bachet in 1621 became the first Latin edition that was widely available. Pierre de Fermat owned a copy, studied it and made notes in the margins. A later 1895 Latin translation by Paul Tannery was said to be an improvement by Thomas L. Heath, who used it in the 1910 second edition of his English translation.

Margin-writing by Fermat and Chortasmenos

The 1621 edition of Arithmetica by Bachet gained fame after Pierre de Fermat wrote his famous "Last Theorem" in the margins of his copy: "If an integer $x$ is greater than 2, then $n$ has no solutions in non-zero integers $an + bn = cn$, $a$, and $b$. I have a truly marvelous proof of this proposition which this margin is too narrow to contain."Fermat's proof was never found, and the problem of finding a proof for the theorem went unsolved for centuries. A proof was finally found in 1994 by Andrew Wiles after working on it for seven years. It is believed that Fermat did not actually have the proof he claimed to have. Although the original copy in which Fermat wrote this is lost today, Fermat's son edited the next edition of Diophantus, published in 1670. Even though the text is otherwise inferior to the 1621 edition, Fermat's annotations—including the "Last Theorem"—were printed in this version.

Fermat was not the first mathematician so moved to write in his own marginal notes to Diophantus; the Byzantine scholar John Chortasmenos (1370–1437) had written "Thy soul, Diophantus, be with Satan because of the difficulty of your other theorems and particularly of the present theorem" next to the same problem.

Other works
Diophantus wrote several other books besides Arithmetica, but only a few of them have survived.

The Porisms
Diophantus himself refers to a work which consists of a collection of lemmas called The Porisms (or Porismata), but this book is entirely lost.

Although The Porisms is lost, we know three lemmas contained there, since Diophantus refers to them in the Arithmetica. One lemma states that the difference of the cubes of two rational numbers is equal to the sum of the cubes of two other rational numbers, i.e. given any $c$ and $a$, with $b$, there exist $a > b$, all positive and rational, such that

Polygonal numbers and geometric elements
Diophantus is also known to have written on polygonal numbers, a topic of great interest to Pythagoras and Pythagoreans. Fragments of a book dealing with polygonal numbers are extant.

A book called Preliminaries to the Geometric Elements has been traditionally attributed to Hero of Alexandria. It has been studied recently by Wilbur Knorr, who suggested that the attribution to Hero is incorrect, and that the true author is Diophantus.

Influence
Diophantus' work has had a large influence in history. Editions of Arithmetica exerted a profound influence on the development of algebra in Europe in the late sixteenth and through the 17th and 18th centuries. Diophantus and his works also influenced Arab mathematics and were of great fame among Arab mathematicians. Diophantus' work created a foundation for work on algebra and in fact much of advanced mathematics is based on algebra. How much he affected India is a matter of debate.

Diophantus has been considered "the father of algebra" because of his contributions to number theory, mathematical notations and the earliest known use of syncopated notation in his book series Arithmetica. However this is usually debated, because Al-Khwarizmi was also given the title as "the father of algebra", nevertheless both mathematicians were responsible for paving the way for algebra today.

Diophantine analysis
Today, Diophantine analysis is the area of study where integer (whole-number) solutions are sought for equations, and Diophantine equations are polynomial equations with integer coefficients to which only integer solutions are sought. It is usually rather difficult to tell whether a given Diophantine equation is solvable. Most of the problems in Arithmetica lead to quadratic equations. Diophantus looked at 3 different types of quadratic equations: $c and d$, $a3 − b3 = c3 + d3$, and $ax2 + bx = c$. The reason why there were three cases to Diophantus, while today we have only one case, is that he did not have any notion for zero and he avoided negative coefficients by considering the given numbers $ax2 = bx + c$, $ax2 + c = bx$, $a$ to all be positive in each of the three cases above. Diophantus was always satisfied with a rational solution and did not require a whole number which means he accepted fractions as solutions to his problems. Diophantus considered negative or irrational square root solutions "useless", "meaningless", and even "absurd". To give one specific example, he calls the equation $b$ 'absurd' because it would lead to a negative value for $c$. One solution was all he looked for in a quadratic equation. There is no evidence that suggests Diophantus even realized that there could be two solutions to a quadratic equation. He also considered simultaneous quadratic equations.

Mathematical notation
Diophantus made important advances in mathematical notation, becoming the first person known to use algebraic notation and symbolism. Before him everyone wrote out equations completely. Diophantus introduced an algebraic symbolism that used an abridged notation for frequently occurring operations, and an abbreviation for the unknown and for the powers of the unknown. Mathematical historian Kurt Vogel states: "The symbolism that Diophantus introduced for the first time, and undoubtedly devised himself, provided a short and readily comprehensible means of expressing an equation... Since an abbreviation is also employed for the word 'equals', Diophantus took a fundamental step from verbal algebra towards symbolic algebra."Although Diophantus made important advances in symbolism, he still lacked the necessary notation to express more general methods. This caused his work to be more concerned with particular problems rather than general situations. Some of the limitations of Diophantus' notation are that he only had notation for one unknown and, when problems involved more than a single unknown, Diophantus was reduced to expressing "first unknown", "second unknown", etc. in words. He also lacked a symbol for a general number $4 = 4x + 20$. Where we would write $x$, Diophantus has to resort to constructions like: "... a sixfold number increased by twelve, which is divided by the difference by which the square of the number exceeds three". Algebra still had a long way to go before very general problems could be written down and solved succinctly.

title: Diophantus