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Cyril M. Kornbluth

Cyril M. Kornbluth (July 2, 1923 – March 21, 1958) was an American science fiction author and a member of the Futurians. He used a variety of pen-names, including Cecil Corwin, S. D. Gottesman, Edward J. Bellin, Kenneth Falconer, Walter C. Davies, Simon Eisner, Jordan Park, Arthur Cooke, Paul Dennis Lavond, and Scott Mariner.

Biography
Kornbluth was born and grew up in the uptown Manhattan neighborhood of Inwood, in New York City. He was of Polish-Jewish descent, the son of a World War I veteran and grandson of a tailor, a Jewish immigrant from Galicia.

The "M" in Kornbluth's name may have been in tribute to his wife, Mary Byers; Kornbluth's colleague and collaborator Frederik Pohl confirmed Kornbluth's lack of any actual middle name in at least one interview.

According to his widow, Kornbluth was a "precocious child", learning to read by the age of three and writing his own stories by the time he was seven. He graduated from high school at thirteen, received a CCNY scholarship at fourteen, and was "thrown out for leading a student strike" without graduating.

As a teenager, he became a member of the Futurians, an influential group of science fiction fans and writers. While a member of the Futurians, he met and became friends with Frederik Pohl, Donald A. Wollheim, Robert A. W. Lowndes, and his future wife Mary Byers. He also participated in the Fantasy Amateur Press Association.

Kornbluth served in the US Army during World War II (European 'Theatre'). He received a Bronze Star for his service in the Battle of the Bulge, where he served as a member of a heavy machine gun crew. Upon his discharge, he returned to finish his education at the University of Chicago under the G.I. Bill. While living in Chicago he also worked at Trans-Radio Press, a news wire service. In 1951 he started writing full-time, returning to the East Coast where he collaborated on novels with his old Futurian friends Frederik Pohl and Judith Merril.

Work
Kornbluth began writing at 15. His first solo story, "The Rocket of 1955", was published in Richard Wilson's fanzine Escape (Vol. 1, No 2, August 1939); his first collaboration, "Stepsons of Mars," written with Richard Wilson and published under the name "Ivar Towers", appeared in the April 1940 Astonishing. His other short fiction includes "The Little Black Bag", "The Marching Morons", "The Altar at Midnight", "MS. Found in a Chinese Fortune Cookie", "Gomez" and "The Advent on Channel Twelve".

"The Little Black Bag" was first adapted for television live on the television show Tales of Tomorrow on May 30, 1952. It was later adapted for television by the BBC in 1969 for its Out of the Unknown series. In 1970, the same story was adapted by Rod Serling for an episode of his Night Gallery series. This dramatization starred Burgess Meredith as the alcoholic Dr. William Fall, who had long lost his doctor's license and become a homeless alcoholic. He finds a bag containing advanced medical technology from the future, which, after an unsuccessful attempt to pawn it, he uses benevolently.

"The Marching Morons" is a look at a far future in which the world's population consists of five billion idiots and a few million geniuses – the precarious minority of the "elite" working desperately to keep things running behind the scenes. In his introduction to The Best of C. M. Kornbluth, Pohl states that "The Marching Morons" is a direct sequel to "The Little Black Bag": it is easy to miss this, as "Bag" is set in the contemporary present while "Morons" takes place several centuries from now, and there is no character who appears in both stories. The titular black bag in the first story is actually an artifact from the time period of "The Marching Morons": a medical kit filled with self-driven instruments enabling a far-future moron to "play doctor". A future Earth similar to "The Marching Morons" – a civilisation of morons protected by a small minority of hidden geniuses – is used again in the final stages of Kornbluth & Pohl's Search the Sky.

"MS. Found in a Chinese Fortune Cookie" (1957) is supposedly written by Kornbluth using notes by "Cecil Corwin", who has been declared insane and incarcerated, and who smuggles out in fortune cookies the ultimate secret of life. This fate is said to be Kornbluth's response to the unauthorized publication of "Mask of Demeter" (as by "Corwin" and "Martin Pearson" (Donald A. Wollheim)) in Wollheim's anthology Prize Science Fiction in 1953.

Biographer Mark Rich describes the 1958 story "Two Dooms" as one of several stories which are "concern[ed] with the ethics of theoretical science" and which "explore moral quandaries of the atomic age":

Many of Kornbluth's novels were written as collaborations: either with Judith Merril (using the pseudonym Cyril Judd), or with Frederik Pohl. These include Gladiator-At-Law and The Space Merchants. The Space Merchants contributed significantly to the maturing and to the wider academic respectability of the science fiction genre, not only in America but also in Europe. Kornbluth also wrote several novels under his own name, including The Syndic and Not This August.

Death
Kornbluth died at age 34 in Levittown, New York. On a day when he was due to meet with Bob Mills in New York City to interview for the position of editor of The Magazine of Fantasy & Science Fiction, he was delayed because he had to shovel snow from his driveway. After running to meet his train following this delay, Kornbluth suffered a fatal heart attack on the platform of the station.

A number of short stories remained unfinished at Kornbluth's death; these were eventually completed and published by Pohl. One of these stories, "The Meeting" (The Magazine of Fantasy & Science Fiction, November 1972), was the co-winner of the 1973 Hugo Award for Best Short Story; it tied with R. A. Lafferty's "Eurema's Dam." Almost all of Kornbluth's solo SF stories have been collected as His Share of Glory: The Complete Short Science Fiction of C. M. Kornbluth (NESFA Press, 1997).

Personality and habits
Frederik Pohl, in his autobiography The Way the Future Was, Damon Knight, in his memoir The Futurians, and Isaac Asimov, in his memoirs In Memory Yet Green and I. Asimov: A Memoir, all give descriptions of Kornbluth as a man of odd personal habits and eccentricities.

Kornbluth, for example, decided to educate himself by reading his way through an entire encyclopedia from A to Z; in the course of this effort, he acquired a great deal of esoteric knowledge that found its way into his stories, in alphabetical order by subject. When Kornbluth wrote a story that mentioned the ballista, an Ancient Roman weapon, Pohl knew that Kornbluth had finished the 'A's and had started on the 'B's.

According to Pohl, Kornbluth never brushed his teeth, and they were literally green. Deeply embarrassed by this, Kornbluth developed the habit of holding his hand in front of his mouth when speaking.

Novels

 * Outpost Mars (1952) (with Judith Merril, writing as Cyril Judd), first published as a Galaxy serial entitled Mars Child (May–July 1951) and later reprinted as Galaxy novel No. 46 retitled Sin in Space (1961)
 * Gunner Cade (1952) (with Judith Merril, writing as Cyril Judd), first published as an Astounding Science Fiction serial (March–May 1952)
 * Takeoff (May 1952), later serialised in New Worlds (April–June 1954)
 * The Space Merchants (April 1953) (with Frederik Pohl), first published as a Galaxy serial entitled Gravy Planet (June–August 1952)
 * The Syndic (October 1953), later serialised in Science Fiction Adventures (December 1953-March 1954), entered the Prometheus Award Hall of Fame in 1986
 * Search the Sky (February 1954) (with Frederik Pohl), later revised by Pohl (October 1985)
 * Gladiator at Law (May 1955) (with Frederik Pohl), first published as a Galaxy serial (June–August 1954), later revised by Pohl (April 1986)
 * Not This August (July 1955) (AKA Christmas Eve), later revised by Pohl (December 1981)
 * Wolfbane (September 1959) (with Frederik Pohl), first published as a Galaxy serial (October–November 1957), later revised by Pohl (June 1986)

Collections

 * The Explorers (1954)
 * "Foreword", [Frederik Pohl]
 * "Gomez"
 * "The Mindworm", 1950
 * "The Rocket of 1955", 1939
 * "The Altar at Midnight", 1952
 * "Thirteen O’Clock" [as by Cecil Corwin], (1941) "Peter Packer" series
 * "The Goodly Creatures", 1952
 * "Friend to Man", 1951
 * "With These Hands", 1951
 * "That Share of Glory", 1962
 * The Mindworm and Other Stories (1955)
 * "The Mindworm”, (1950)
 * "Gomez”, 1954
 * "The Rocket of 1955”, 1939
 * "The Altar at Midnight”, 1952
 * "The Little Black Bag”, 1950
 * "The Goodly Creatures”, 1952
 * "Friend to Man”, 1951
 * "With These Hands”, 1951
 * "That Share of Glory”, 1952
 * "The Luckiest Man in Denv" [as by Simon Eisner], · 1952
 * "The Silly Season”, 1950
 * "The Marching Morons · nv Galaxy Apr ’51
 * A Mile Beyond the Moon (1958) [abridged for its 1962 paperback reprint, see below]
 * "Make Mine Mars”, 1952
 * "The Meddlers”, 1953 [not in 1962 paperback]
 * "The Events Leading Down to the Tragedy”, 1958
 * "The Little Black Bag”, 1950 (related to "The Marching Morons")
 * "Everybody Knows Joe”, 1953
 * "Time Bum”, 1953
 * "Passion Pills”, [original here] [not in 1962 paperback]
 * "Virginia”, 1958
 * "The Slave”, 1957 [not in 1962 paperback]
 * "Kazam Collects" [as by S. D. Gottesman], 1941
 * "The Last Man Left in the Bar”, 1947 – "a confrontation between aliens and a magnetron technician, written with an audacious literary command"
 * "The Adventurer”, 1953
 * "The Words of Guru" [as by Kenneth Falconer], 1941 – "an early but striking fantasy about a genius child acquiring supernatural power"
 * "Shark Ship" ["Reap the Dark Tide"], 1958
 * "Two Dooms”, 1958 [not in 1962 paperback]
 * The Marching Morons (and other Science Fiction Stories) (1959)
 * "The Marching Morons”, 1951
 * "Dominoes”, 1953
 * "The Luckiest Man in Denv" [as by Simon Eisner], 1952
 * "The Silly Season”, 1950
 * "MS. Found in a Chinese Fortune Cookie”, 1957
 * "The Only Thing We Learn”, 1949
 * "The Cosmic Charge Account”, 1956
 * "I Never Ask No Favors”, 1954
 * "The Remorseful”, 1953
 * The Wonder Effect (1962) (with Frederik Pohl)
 * "Introduction”,
 * "Critical Mass”, 1962
 * "A Gentle Dying”, 1961
 * "Nightmare with Zeppelins", 1958
 * "Best Friend" [as by S. D. Gottesman], 1941
 * "The World of Myrion Flowers”, 1961
 * "Trouble in Time" [as by S. D. Gottesman], 1940
 * "The Engineer”, 1956
 * "Mars-Tube [as by S. D. Gottesman]”, 1941
 * "The Quaker Cannon”, 1961
 * Best Science Fiction Stories of C. M. Kornbluth (1968)
 * "Introduction”, [Edmund Crispin]
 * "The Unfortunate Topologist”, 1957 (poem)
 * "The Marching Morons”, 1951
 * "The Altar at Midnight”, 1952
 * "The Little Black Bag”, 1950
 * "The Mindworm”, 1950
 * "The Silly Season”, 1950
 * "I Never Ask No Favors”, 1954
 * "Friend to Man”, 1951
 * "The Only Thing We Learn”, 1949
 * "Gomez”, 1954
 * "With These Hands”, 1951
 * "Theory of Rocketry”, 1958
 * "That Share of Glory”, 1952
 * Thirteen O'Clock and other Zero Hours (1970) (edited by James Blish) stories published originally as by "Cecil Corwin" plus "MS. Found in a Chinese Fortune Cookie" (see above)
 * "Preface”, [James Blish]
 * "Thirteen O’Clock [combined version of the "Peter Packer" stories, “Thirteen O’Clock” and “Mr. Packer Goes to Hell”, both 1941]”, [first combined appearance here]
 * "The Rocket of 1955”, 1939
 * "What Sorghum Says" [as by Cecil Corwin], 1941
 * "Crisis!" [as by Cecil Corwin], 1942
 * "The Reversible Revolutions" [as by Cecil Corwin], 1941
 * "The City in the Sofa" [as by Cecil Corwin], 1941
 * "The Golden Road" [as by Cecil Corwin], 1942
 * "MS. Found in a Chinese Fortune Cookie”, 1957
 * The Best of C. M. Kornbluth (1976)
 * "An Appreciation”, [Frederik Pohl]
 * "The Rocket of 1955”, 1939
 * "The Words of Guru" [as by Kenneth Falconer], 1941
 * "The Only Thing We Learn”, 1949
 * "The Adventurer”, 1953
 * "The Little Black Bag”, 1950
 * "The Luckiest Man in Denv" [as by Simon Eisner], 1952
 * "The Silly Season”, 1950
 * "The Remorseful”, 1953
 * "Gomez”, 1954
 * "The Advent on Channel Twelve”, 1958
 * "The Marching Morons”, 1951
 * "The Last Man Left in the Bar”, 1957
 * "The Mindworm”, 1950
 * "With These Hands”, 1951
 * "Shark Ship" [“Reap the Dark Tide”], 1958
 * "Friend to Man”, 1951
 * "The Altar at Midnight”, 1952
 * "Dominoes”, 1953
 * "Two Dooms”, 1958

Spider Robinson praised this collection, saying "I haven't enjoyed a book so much in years." Mark Rich wrote, "Critics judging Kornbluth by this anthology, edited by Pohl, have seen a growing bitterness in his later stories. This reflects editorial choice more than reality, because Kornbluth also wrote delightful humor in his last years, in stories not collected here. These tales demonstrate Kornbluth's effective use of everyday individuals from a variety of ethnic backgrounds as well as his well-tuned ear for dialect."
 * Critical Mass (1977) (with Frederik Pohl)
 * "Introduction”, (Pohl)
 * "The Quaker Cannon”, 1961
 * "Mute Inglorious Tam”, 1974
 * "The World of Myrion Flowers”, 1961
 * "The Gift of Garigolli”, 1974
 * "A Gentle Dying”, 1961
 * "A Hint of Henbane”, 1961
 * "The Meeting”, 1972
 * "The Engineer”, 1956
 * "Nightmare with Zeppelins”, 1958
 * "Critical Mass”, 1962
 * "Afterword”, (Pohl)
 * Before the Universe (1980) (with Frederik Pohl)
 * "Mars-Tube" [as by S. D. Gottesman (with Frederik Pohl)], 1941
 * "Trouble in Time" [as by S. D. Gottesman (with Frederik Pohl)], 1940
 * "Vacant World" [as by Dirk Wylie (with Dirk Wylie, and Frederik Pohl)], 1940
 * "Best Friend" [as by S. D. Gottesman (with Frederik Pohl)], 1941
 * "Nova Midplane" [as by S. D. Gottesman (with Frederik Pohl)], 1940
 * "The Extrapolated Dimwit" [as by S. D. Gottesman (with Frederik Pohl)], 1942
 * Our Best: The Best of Frederik Pohl and C.M. Kornbluth (1987) (with Frederik Pohl)
 * "Introduction”, (Pohl)
 * "The Stories of the Sixties”, (Pohl, section introduction)
 * "Critical Mass”, 1962
 * "The World of Myrion Flowers”, 1961
 * "The Engineer”, 1956
 * "A Gentle Dying”, 1961
 * "Nightmare with Zeppelins”, 1958
 * "The Quaker Cannon”, 1961
 * "The 60/40 Stories”, (Pohl, section introduction)
 * "Trouble in Time" [as by S. D. Gottesman], 1940
 * "Mars-Tube" [as by S. D. Gottesman], 1941
 * "Epilogue to The Space Merchants”, (Pohl, section introduction)
 * "Gravy Planet”, (extract from the magazine serial, not used in the book)
 * "The Final Stories”, (Pohl, section introduction)
 * "Mute Inglorious Tam”, 1974
 * "The Gift of Garigolli”, 1974
 * "The Meeting”, 1972
 * "Afterword”, (Pohl)
 * His Share of Glory: The Complete Short Science Fiction of C.M. Kornbluth (1997) – this includes almost all of Kornbluth's solo fiction, but does not include all of the collaborative pseudonymous works which were published among his earliest work between 1940 and 1942, some of which were published in Before the Universe (1980).
 * "Cyril”, [Frederik Pohl]
 * "Editor’s Introduction”, [Timothy P. Szczesuil]
 * "That Share of Glory”, 1952
 * "The Adventurer”, 1952
 * "Dominoes”, 1953
 * "The Golden Road" [as by Cecil Corwin], 1942
 * "The Rocket of 1955”, 1939
 * "The Mindworm”, 1950
 * "The Education of Tigress McCardle”, 1957
 * "Shark Ship" [“Reap the Dark Tide”], 1958
 * "The Meddlers”, 1953
 * "The Luckiest Man in Denv" [as by Simon Eisner], 1952
 * "The Reversible Revolutions [as by Cecil Corwin], 1941
 * "The City in the Sofa" [as by Cecil Corwin], 1941
 * "Gomez”, 1954
 * "Masquerade" [as by Kenneth Falconer], 1942
 * "The Slave”, 1957
 * "The Words of Guru" [as by Kenneth Falconer], 1941
 * "Thirteen O’Clock" [as by Cecil Corwin], 1941
 * "Mr. Packer Goes to Hell" [as by Cecil Corwin], 1941
 * "With These Hands”, 1951
 * "Iteration”, 1950
 * "The Goodly Creatures”, 1952
 * "Time Bum”, 1953
 * "Two Dooms”, 1958
 * "Passion Pills”, 1958
 * "The Silly Season”, 1950
 * "Fire-Power" [as by S. D. Gottesman], 1941
 * "The Perfect Invasion" [as by S. D. Gottesman], 1942
 * "The Adventurers”, 1955
 * "Kazam Collects" [as by S. D. Gottesman], 1941
 * "The Marching Morons”, 1951
 * "The Altar at Midnight”, 1952
 * "Crisis!" [as by Cecil Corwin], 1942
 * "Theory of Rocketry”, 1958
 * "The Cosmic Charge Account”, 1956
 * "Friend to Man”, 1951
 * "I Never Ast No Favors”, 1954
 * "The Little Black Bag”, 1950
 * "What Sorghum Says" [as by Cecil Corwin], 1941
 * "MS. Found in a Chinese Fortune Cookie”, 1957
 * "The Only Thing We Learn”, 1949
 * "The Last Man Left in the Bar”, 1957
 * "Virginia”, 1958
 * "The Advent on Channel Twelve”, 1958
 * "Make Mine Mars”, 1952
 * "Everybody Knows Joe”, 1953
 * "The Remorseful”, 1953
 * "Sir Mallory’s Magnitude" [as by S. D. Gottesman], 1941
 * "The Events Leading Down to the Tragedy”, 1958
 * "King Cole of Pluto" [as by S. D. Gottesman], 1940
 * "No Place to Go" [as by Edward J. Bellin], 1941
 * "Dimension of Darkness" [as by S. D. Gottesman], 1941
 * "Dead Center" [as by S. D. Gottesman], 1941
 * "Interference" [as by Walter C. Davies], 1941
 * "Forgotten Tongue" [as by Walter C. Davies], 1941
 * "Return from M-15" [as by S. D. Gottesman], 1941
 * "The Core" [as by S. D. Gottesman], 1942

Non-science fiction

 * The Naked Storm (1952, as Simon Eisner)
 * Valerie (1953, as Jordan Park), a novel about a girl accused of witchcraft
 * Half (1953, as Jordan Park), a novel about an intersex person
 * A Town Is Drowning (1955, with Frederik Pohl)
 * Presidential Year (1956, with Frederik Pohl)
 * Sorority House (1956, with Frederik Pohl, as Jordan Park), a lesbian pulp novel
 * A Man of Cold Rages (1958, as Jordan Park), a novel about an ex-dictator

Uncollected short stories

 * "Stepsons of Mars", (1940) [as "Ivar Towers" (with Richard Wilson)
 * "Callistan Tomb", (1941) [as "Paul Dennis Lavond" (with Frederik Pohl)]
 * "The Psychological Regulator", (1941) [as "Arthur Cooke" (with Elsie Balter {later Elsie Wollheim}, Robert A. W. Lowndes, John Michel, Donald A. Wollheim)
 * "The Martians Are Coming", (1941) [as "Robert A W Lowndes" (with Robert A. W. Lowndes)]
 * "Exiles of New Planet", (1941) [as "Paul Dennis Lavond" (with Frederik Pohl, Robert A. W. Lowndes, Dirk Wylie)]
 * "The Castle on the Outerplanet", (1941) [as "S D Gottesman" (with Frederik Pohl, Robert A. W. Lowndes)]
 * "A Prince of Pluto", (1941) [as "S D Gottesman" (with Frederik Pohl)]
 * "Einstein's Planetoid", (1941) [as "Paul Dennis Lavond" (with Frederik Pohl, Robert A. W. Lowndes, Dirk Wylie)]
 * "An Old Neptunian Custom", (1942) [as "Scott Mariner" (with Frederik Pohl)]

Articles

 * "A Funny Article on the Convention", (1939)
 * "New Directions", (1941) [as "Walter C. Davies"]
 * "The Failure of the Science Fiction Novel as Social Criticism", in The Science Fiction Novel: Imagination and Social Criticism, ed. Basil Davenport, Advent Press, 1959. (pages 64–101). Brian Stableford called this "an important early piece of sf criticism, sharply pointing out the genre's shortcomings."

Trivia
Kornbluth's name is mentioned in Lemony Snicket's Series of Unfortunate Events as a member of V.F.D., a secret organization dedicated to the promotion of literacy, classical learning, and crime prevention.

title: Cyril M. Kornbluth


text:
Talk:Chełmno Land

Old Talk
I changed 'sectioned' to 'divided' and 'bishoprics' to 'dioceses' (both more common English usages). The huge pile of names in the last part of the article is impenetrable. I suppose that 'border to' should be 'border of'. --User:MichaelTinkler 17:51, 25 February 2002

 * I have changed spelling of this area's name. Kulmerland is German name only. It has nothing to do with English. -- User:Von Fiszman (Talk | contribs) at 16:48, 11 January 2004.

1. Polish: Ziemia Chelminska, German: Kulmerland, English: Culmland. -er' does not exist in English. English name in English Wikipedia, please.

2. Chelmno Land was settled by Lekhitic tribes long time before any province of Poland was created. It is proven by archeological researches (Old Prussian settlements started about 20 kilometers north of Osa river in 10th century. Later (10th-13th centuries) Polish settlement moved even more north. Almost all names in the area are of Slavonic origins, in contrary to former Gallindia or Pomesania. You do not have to be a scientist to check it.

3. Chelmno Land is not a part of Prussia, because; a) there is no land named Prussia at present. b) was not inhabited by Old Prussians, at least in historical times. c) historically it is a separate region.

4. ''Since the 20th century conquest it is a part of Kuyavian-Pomeranian Voivodship. '' - What are you writing about?

5. Kulm is German. I do not see any reason why for the Polish town would be used German name in English Wikipedia.

6. The place names comprising 'cholm element are very common in Poland. Latin name in the middle of Central Europe? 1000 kilometers from Romanic languages area. It is creazy.

7. The following text is a Polish wikipedia entry.

I have written at least the first part of the entry. Belive me, it is not from Polish Wikipedia.

8. English names: Chelmno Land (or Culmland or Culm Land), Warmia, Sambia. Leave German names for German Wikipedia.

9. Land belonging to Poland till 1228 was undoubtelly recovered in 1466. I suppose that both understand the mening of the word.

Regards, -- User:Yeti 00:04, 25 February 2004

Old Map of Culmerland in Altes Preussenland - Ancient Prussia
Old Map of Culmerland/Kulmerland in Prussia

Wikipedia article Chelmno Land as of Feb, Mar 2006 is completely oneside POV. It does not portrait factual history. MG 2/2/2006, 3/10/2006 —Preceding unsigned comment added by 24.23.39.36 (talk • contribs) 00:08, 10 March 2006

 * To improve the article, it would help if you listed here which text you disagree with. Olessi 18:25, 10 March 2006 (UTC)

Maps of areas mentioned..
I am a visual person. Can we have the ancient and modern maps of Chelmno Land as well as a map showing the location within Europe. Thanks --Jim Jacobs 05:18, 10 April 2006 (UTC)

Name?
In was wondering about the current name "Chełmno Land" which according to post 1990- Google Books hits is not used at all:
 * "Chełmno Land" date:1990-2008 0
 * "Chelmno Land" date:1990-2008 2
 * "District of Chelmno" date:1990-2008 3

This compares to
 * "Kulmerland" date:1990-2008 54
 * "Kulmerland" -Komet -ship -Schiff -sea date:1990-2008 41 (excluding uses that point to Kulmerland (ship) of WW2)
 * Kulmerland" +the -Komet -ship -Schiff -sea date:1990-2008 15 (excluding uses that point to Kulmerland (ship) of WW2) enforcing English use
 * Kulmerland" +das -Komet -ship -Schiff -sea date:1990-2008 26 (excluding uses that point to Kulmerland (ship) of WW2) enforcing German use
 * "Culmerland" date:1990-2008 7
 * "Kulmer Land" date:1990-2008 78 (mostly in German)
 * "Kulm land" date:1990-2008 0
 * "Culm land" date:1990-2008 0
 * "Culmer Land" date:1990-2008 1
 * "ziemia chełmińska" date:1990-2008 19
 * "ziemia chelminska" date:1990-2008 4

Thus, I suggest a move to "Kulmerland", as used in English, among others on Google Scholar, by -- Matthead Discuß   20:37, 25 May 2008 (UTC)
 * Medieval Missionary: A Study of the Conversion of Northern Europe Ad 500 to 1300, by James Thayer Addison, 2003
 * The Close of the Middle Ages 1273 to 1494 V3, by Sir R. Lodge - History - 2005
 * Castles of the Western World: With 240 Illustrations, by Armin Tuulse - Architecture - 2002
 * Social System, Rationality and Revolution, by Leszek Nowak, Marcin Paprzycki - Social Science - 1993
 * Encyclopedia of the Middle Ages, by André Vauchez, Richard Barrie Dobson, Michael Lapidge - History - 2001
 * Teutonic Knight by David Nicolle, Graham Turner - History - 2007 - 64 pages

 * I am not convinced; your above numbers don't exclude German language publications for Kulmerland, for example.--Piotr Konieczny aka Prokonsul Piotrus 23:44, 25 May 2008 (UTC)
 * Does not convinced also apply for the current title? I have listed 6 clearly English books above, taken from the first page with 10 listed, of the above 41 hits for the non-floating "Kulmerland". And there are also the hits for "Culmerland", almost all in English. Anyway, English use can be easily enforced with "the", which yields 16 hits compared to 26 German (by "das"). So more modern books on Kulmerland are published in German rather than in English (or Polish), hardly surprising. Besides, does that support the current title in any way, which still stands at 0 with the diacritic, and 2 without? And these two are odd mixtures of German and English published in Poland - as is this edit. Reminds also of the article created under "Chełmno law" instead the proper Kulm law. The land was and is defined by the deal between Conrad and the Order, and subsequent 700+ years of German settlement. -- Matthead Discuß   02:16, 26 May 2008 (UTC)
 * May I suggest a WP:RM? --Piotr Konieczny aka Prokonsul Piotrus 22:11, 26 May 2008 (UTC)
 * May I suggest doing as you did several times elsewhere - moving, and salting the redirect to prevent restoration? -- Matthead Discuß   22:48, 26 May 2008 (UTC)
 * Done. -- Matthead Discuß   20:39, 3 August 2008 (UTC)
 * Matthead, may I ask you to revert this rename and file a proper WP:RM instead, as Piotrus suggested above ? --Lysytalk 07:36, 5 August 2008 (UTC)

The google hits presented by Matthead are related not to Chelmno land but to a German ship named "Kulmerland". --Lysytalk 06:50, 5 August 2008 (UTC)

Requested move

 * The following discussion is an archived discussion of the proposal. Please do not modify it. Subsequent comments should be made in a new section on the talk page. No further edits should be made to this section.

The result of the proposal was no consensus to support move. JPG-GR (talk) 03:11, 21 August 2008 (UTC)

Kulmerland → Province of Culm or Culmer Land or Culm land— most common spellings in English for the historic German period of the province.--Termer (talk) 05:39, 20 August 2008 (UTC)

Survey

 * Feel free to state your position on the renaming proposal by beginning a new line in this section with  or  , then sign your comment with  . Since polling is not a substitute for discussion, please explain your reasons, taking into account Wikipedia's naming conventions.

 * Object. Kulmerland (Polish administrative region of ziemia) is not the same as its bishopric.--Piotr Konieczny aka Prokonsul Piotrus 18:03, 19 August 2008 (UTC)

 * Comment: This article seems to be not about a Polish administrative region but about the German period. The bottom line, the title should be spelled in English and I have already suggested an alternative since the bishopric was only a part of the historic German province, other parts were administrated by the Teutonic knights. --Termer (talk) 05:36, 20 August 2008 (UTC)

PS. renaming the article that speaks about the German period Culmerland, Culm land might work even better than Province of Culm currently suggested.--Termer (talk) 05:36, 20 August 2008 (UTC)

Discussion

 * Any additional comments:

There already exists Bishopric of Culm (Chełmno), although I don't see the need to include double naming in that title. Olessi (talk) 15:53, 9 August 2008 (UTC)
 * Are you saying that the articles should be merged? The current article title clearly violates WP:NPOV since the place is known by 2 different names given by either the German or Polish rule. Changing it to an administrative unit would solve the problem. And I noticed there are already Chełmno Voivodeship and Chełmno County so the Polish periods are covered.--Termer (talk) 17:45, 9 August 2008 (UTC)

Actually, one of the most common names in English for the region that was ruled by the German Bishopric of Culm (Chełmno) and Teutonic knights would be Province of Culm (45 returns), and there is also one Province of Kulmerland. Should we go for Province of Culm (Kulmerland ziemia chełmińska) to cover the German era of the region?--Termer (talk) 18:10, 9 August 2008 (UTC)

Here is my understanding of the territory's historical progression:
 * "Chełmno Land", battleground between Poles and Prussians from the 11th-13th centuries
 * "Kulmerland", the launching point of the Teutonic Knights' invasion of Prussia beginning in the 1230s
 * Bishopric of Culm, created in 1243. The bishops only received a third of the Kulmerland. Like the bishops of Samland and Pomesania, the bishops of Culm were more subordinated to the Knights than the Bishops of Warmia. The rest of the Kulmerland territory was divided amongst Komturei (i.e. Strasburg, Brattian) run by Teutonic Knights.
 * Chełmno Voivodeship/Palatinate, subdivision of Royal Prussia within Poland/Polish-Lithuanian Commonwealth from 1466. Included Chełmno County.
 * Kreis Culm, district of Prussian West Prussia. Formed in 1772. Since Thorn/Toruń was not annexed by Prussia until 1793, I presume the Chełmno Voivodeship lasted until 1793/95, even though Kulm/Chełmno was annexed by Prussia in 1772.
 * Bydgoszcz Department, part of the Duchy of Warsaw (1807-1815), after which it reverted to Prussian West Prussia
 * Chełmno County, part of the Pomeranian Voivodeship (1919-1939) of the Second Polish Republic
 * Annexed into Nazi Germany as part of Reichsgau Danzig-West Prussia in 1939. The town was included within Landkreis Kulm (renamed Landkreis Kulm (Weichsel) in 1941).
 * Included within Poland's Bydgoszcz Voivodeship (1946-1975), Toruń Voivodeship (1975-1988), Kuyavian-Pomeranian Voivodeship (1988-).

Based on that summary, a suggestion would be to have the Kulmerland article focus on the period until 1466, including the Bishopric of Culm and the Komturei. Olessi (talk) 19:30, 9 August 2008 (UTC)

 * And the title for the article including the Bishopric of Culm and the Komturei would be...? I'd suggest going for Province of Culm or Culmer Land like in similar cases, former German provinces in the region Kurland, Estland, Livland etc. are using the most common English names on WP instead of historic and/or traditional German versions.--Termer (talk) 20:55, 9 August 2008 (UTC)

PS. Shouldn't Komturei be in the context Commandry? I think Komturei refers to a rank -Commandries of the Teutonic Knights versus Commandry, that was a regional unit.--Termer (talk) 21:08, 9 August 2008 (UTC)
 * The above discussion is preserved as an archive of the proposal. Please do not modify it. Subsequent comments should be made in a new section on this talk page. No further edits should be made to this section.

Kulmerland, Chelmno land
Apparently the article was moved based on this discussion above, which was used as a pretext for moving the article and salting the redirect (so that it couldn't be moved back). It also seems that most of the hits for the German name "Kulmerland" were for a ... ship, rather than a place.

Most strikingly though is that the claim that there is 0 hits for "Chelmno land" is completely false, as evidenced by the very gbook link provided - in fact that very link shows 63 hits.

Now, ok, I understand that if we're talking about the part of history, 1226–1466, when this place was a Teutonic province then Kulmerland might be the appropriate title. However, currently the article is totally schizophrenic. The text is obviously about more than just this place's time as a Teutonic province - the history goes from 1065 until 1945 (and should probably go further into near past). Yet the infobox, which I'm about to delete since it is not appropriate for THIS article, is about the 1226–1466 period.

So what I think should be done is that a proper "Kulmerland" article on the Teutonic province for 1226-1446 should be created (though the title Kulmerland (Teutonic province) might be better, while this more general article should be moved (ugh, manually) back to "Chelmno land".Volunteer Marek (talk) 20:27, 15 April 2011 (UTC)

Btw, if an article starts with "x is a German name for..." then that pretty clearly indicates that the article title is inappropriate. We don't have an article for Dalmatiner which says "Dalmatiner is a German name for a Dalmatian..." either, just a redirect to "Dalmatian". This is the English wikipedia.Volunteer Marek (talk) 20:34, 15 April 2011 (UTC)
 * If you have time to rewrite/split the article, great. I think your split proposal makes sense. Alternatively, looking at the pl wiki, note that the article there covers the territory history up to present. If we mirror this, then the article could be moved to the Polish name, as it would no longer be about historical, but present entity. Within the article, of course, it would make sense to have a section entitled Kulmerland, about the period it was a Teutonic province. --Piotr Konieczny aka Prokonsul Piotrus| talk 20:14, 19 July 2011 (UTC)
 * Well, this article already does cover the history of the region from 1050 or so until 1945. The bogus nature of this article can also been seen in the first sentence: "Kulmerland is the German name for ...". It's basically as if someone moved the article on New England to Neuengland and changed the first sentence of that article to "Neuengland is the German name for..." So the fundamental problem here is getting the article moved back to the name it should be under and fixing the disruptive salting of the previous name.Volunteer Marek (talk) 09:41, 20 July 2011 (UTC)

 * Also, note that the part of the article which is about the Teutonic Knights is only three sentences long.Volunteer Marek (talk) 10:02, 20 July 2011 (UTC)

Much of history is lacking
Just by inspecting the dates of the events given in the current version, the region's history between 1200 and 1900 seems underrepresented. In particular, the region's German history should be treated in a neutral manner. — Preceding unsigned comment added by Stan Tincon (talk • contribs) 15:11, 2 April 2018 (UTC)

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

title: Talk:Chełmno Land


text:
Talk:Chinese remainder theorem

Algebraic method
has recently added a section to the article. I have reverted it with the edit summary "Unsourced, and the general description of the method is lacking". Fly by Night has reinstalled this section (with a reference added) after qualifying (on my talk page) my revert of unfair. By this he violates both WP:NPA and WP:BRD rules: WP:BRD states clearly that, when an edit is reverted, one should not revert the revert. Instead, one has to discuss on the talk page for searching for a consensus. Therefore, I will revert again Fly by Night's edit. Please do not reinsert it without discussing the content here, and raising the following points. D.Lazard (talk) 16:07, 31 August 2016 (UTC)
 * Heading: This method is no more algebraic than the others given in the other sections. Thus qualifying it of "algebraic" is confusing for the layman, and must be avoided.
 * Heading again: Although the heading contains "method", I do not see any method in the section, only ad hoc computation on a specific example.
 * Content: As a professional mathematician, I am unable to understand the method behind this computation.Thus, there is not hope that a normal reader can use it for running another example.
 * Content: It seems that the method consists in applying the method, but without citing it and without saying how it has to be modified for dealing with the fact that one want only integer solutions. This has to be clarified before reinstalling this section.
 * When I describe your edit as unfair, I did not say anything about you personally, so I really can't see how I have made a personal attack. I'm also flabbergasted by your reference to WP:BRD given that you reverted without discussion. I was the one to initiate contact! In reference to your objections: If you don't like the heading then change it. If you don't like the title "algebraic" then suggest a better one. If you think I've applied a method without naming it then name it. Yes, the method referes to a specific example, but so too do the other two. Please, sit down and work through another example - the method is sound. This is a public encyclopedia and we should help one another create better content. The main way to do that is to make improvements that you think necessary. I've looked through your edit history and you often make wholesale deletion of content that you disagree with. This is not fair. I am trying to improve this encyclopedia. Why not work with me and make changes to my edit to improve it? — Fly by Night  (  talk  )  19:20, 1 September 2016 (UTC)
 * My reading of WP:BRD would be that it does not apply to the first revert, but rather to trying to restore changes that have been reverted. As to the addition, I would say it needs heavy improvement. It needs a brief introduction and explanation, it needs the proper referencing (this is a variant of "back substitution" or elimination of variables, as noted), it needs a better title (no more or less algebraic than other sections). It could also benefit from comparisons to the other methods (it tends to work better and faster than the constructive proof when the moduli are small, for example). Given that it requires extensive improvement before it is ready, I think D.Lazard used the cycle described in WP:BRD correctly; at this point, it should be discussed in the talk page so that it can be whipped into shape before being re-added. Just saying "If you think this is not good enough, then fix it!" is to try to shift the responsibility for making a suitable addition where it shouldn't be in the first place. Magidin (talk) 22:12, 1 September 2016 (UTC)
 * I share none of D.Lazard's objections. I have other somewhat related concerns with the article, however: the entire "finding the solution" section is not very well written, nor cited.  The first two methods are essentially the same, as are the latter two methods.  The first two methods are absurd as a practical matter.  The latter two methods are not absurd, but they are redundant with the second proof of existence.  Some trimming and rearranging is definitely in order.  --JBL (talk) 22:16, 1 September 2016 (UTC)
 * If I were to give a completely accurate (but unsourceable) name to this method, it would "intersection of cosets by substitution and modular arithmetic". Each congruence $$x \equiv t \bmod{m}$$ defines a coset of the subgroup mZ, and any technique for solving simultaneous congruences can be interpreted as a method for intersecting these cosets.  The content of the method under discussion appears to be that two cosets can be intersected by substituting a generator for one into the equation defining the other, and then modular arithmetic allows one to solve for an additional condition on the generator.
 * I have one other comment. Suppose one wishes to solve a congruence modulo $$m_1, \ldots, m_k$$.  The so-called "brute force approach" requires computing $$\textstyle\sum_i \prod_{j \neq i} m_j = (\prod_j m_j)(\sum_i m_i^{-1})$$ values.  If all moduli are roughly equal to some constant M, then this is roughly $$kM^{k-1}$$ operations.  The "exhaustive search" requires at most $$\textstyle\sum_i \prod_{j > i} m_j$$ values.  This is roughly $$M^{k-1} + M^{k-2} + \cdots + M$$ operations.  When written in O-notation, the difference between these two gets swallowed up and they are both $$O(M^{k-1})$$.  Nevertheless, since the main interest in these methods is hand computation, I think the article is wrong to say that this method is "essentially the same method" as the "brute force approach".  The constant matters a great deal when you are doing arithmetic by hand.  (Of course, if one is comfortable computing modular inverses by hand, then the other methods are faster still.)  Ozob (talk) 00:20, 2 September 2016 (UTC)

Do you think the content should be included - perhaps subject to some revision? — Fly by Night  (  talk  )  18:23, 2 September 2016 (UTC)
 * I think the content should not be included unless extensively revised. So I support its initial revert. Magidin (talk) 18:30, 2 September 2016 (UTC)
 * Could you please suggest the revisions? The content that I added was an adaptation of a method from a well-known textbook for first year undergraduates. — Fly by Night  (  talk  )  20:27, 2 September 2016 (UTC)
 * I am familiar with the method; I teach it in class. And I already suggested what I think it needs, in broad strokes: "It needs a brief introduction and explanation, it needs the proper referencing (this is a variant of "back substitution" or elimination of variables, as noted), it needs a better title (no more or less algebraic than other sections). It could also benefit from comparisons to the other methods (it tends to work better and faster than the constructive proof when the moduli are small, for example)."  If your question was meant instead to be, "Could you please do the work of improving it for me?" then, no. Magidin (talk) 20:45, 2 September 2016 (UTC)
 * My question was not "Could you please do the work of improving it for me?" That's why I asked you to suggest revisions. Having said that, let's try to remember that this is a cooperative project where we should help one another. Please don't take this the wrong way; try to see it from my point of view: I made a good faith edit which was referenced and which you teach to your own classes but it was simply deleted. There is no way for me to proceed. Someone doesn't like an edit so they delete it. People say that improvements need to be made, but won't help to make them. We find ourselves in a situation where a well-known method that is taught around the world cannot be added to the article because one person doesn't like what was written. — Fly by Night  (  talk  )  22:55, 2 September 2016 (UTC)
 * I believe that this section of the article (not just this particular method) needs wholesale revision. There should be clear statements of the algorithms being proposed; it doesn't have to be overly formal, and I would like to keep the examples, but there should be something more general for someone who doesn't yet grasp the underlying principles.  Ozob (talk) 23:52, 2 September 2016 (UTC)
 * I agree. But how do I go about making such revisions/additions when they are immediately deleted? — Fly by Night  (  talk  )  01:32, 3 September 2016 (UTC)

 * You are asking me to make the revisions. Whether you characterize them as "suggestions" or not, you are asking that I do the work of rewriting your material so that it can be added. I think this needs too much revision, and too much effort, that I simply cannot spare the time to do right now (reviewing a proposed addition or making minor changes to it, by contrast, would not take as much time). As part of my cooperation in this, I indicated the type of change and content that I believe the portion would require. You claim this makes it impossible for you to proceed; but that simply means that you are unwilling or unable to engage with the suggestions made ("add an explanation", "add an introduction", "compare it to the other methods"). That is help; when I referee a paper, I don't rewrite it for the author: I tell the author what kind of changes should be made, and let the author attempt them, for example; that does not mean I'm not helping. Your characterization of the situation is also rather disingenuous: I do not know if it is a "well-known method taught around the world" (do you have any evidence for this? I am only aware of it being taught in two countries, and not universally in either; and the method is not known to many people I know). It wasn't just one person who didn't like what was written (I also find the way it was proposed to be insufficient and in need of too much rewriting to warrant inclusion until that is done, for one). It is also patently false that "it cannot be added"; it just should not be added written in the manner in which you attempted to add it. Perhaps cutting down on the hyperbole would help. If you find this frustrating, well, I find your reaction to be frustrating as well. At this point, what I would try to do with an addition is to read the criticisms and suggestions presented, and attempt to incorporate them into a write-up that I would put up in the talk page as a suggestion for addition. If you cannot or will not do that, well, then you'll have to wait until I or other people have the time and inclination to do it for you.Magidin (talk) 01:01, 3 September 2016 (UTC)

 * I'm probably being foolish, but where are the hyperbolae in my addition. (Please read it) — Fly by Night  (  talk  )  01:32, 3 September 2016 (UTC)

 * The hyperbole are in your comments. "There is no way to proceed" (yes, there is); "well-known method taught around the world" (at best an unwarranted pair of assertions); "cannot be added because one person doesn't like it" (no, it wasn't just one person, and there was more expressed than a simple personal dislike). As to how do you go about making the corrections/additions: you make the proposals in the talk page; they will not be "immediately deleted" in the talk page. That's how consensus is built. Oh, and by the way, thanks for the insinuation that I am commenting without having read your addition in the first place. For the record, I read it before I first commented, so thanks for the vote of confidence in my integrity and honesty. Magidin (talk) 03:17, 3 September 2016 (UTC)

Finding the solution
It has been said in the preceding section that the whole section needs a revision. I agree, and will begin to do it, along the following line: D.Lazard (talk) 16:59, 3 September 2016 (UTC)
 * renaming the section "Computation"
 * expanding the general introduction of the section by adding to the example the general specification of the computation problem
 * replacing the content of section "Brute-force approach" by the first method of next subsection (IMO, this brute-force approach has been imagined by a set-theorist who has never computed anything; it is difficult to imagine a method which is together so inefficient and so complicate)
 * adding indication on the complexity of the different methods
 * expanding the section "using the existence construction"
 * adding a section "As a linear Diophantine system": all methods for linear Diophantine systems may be used here, and 's one is only one of them.

 * Thank you! I look forward to it.  Ozob (talk) 18:53, 3 September 2016 (UTC)
 * Indeed, thank you; I wish I had the time. I'll pitch in as time permits. Magidin (talk) 20:24, 3 September 2016 (UTC)

 * I have completed above edit program. Feel free for improving the result. In particular, references to The Art of Computer Programming should be added and used for improving details (I have not this book under hands, and some details of my edits are based on what I remember of my readings). D.Lazard (talk) 10:09, 9 September 2016 (UTC)

 * The mission of Wikipedia is that it be an encyclopedia, but Wikipedia is infamous for being incomprehensible to non-experts. The section on simultaneous Diophantine equations gives no method, and the links are not very helpful. I think that an explicit example, like the one I originally added, would be a good idea. If we want people with basic mathematical literacy to get something from the article then I think we need the example I gave.  I'm sorry, but all of the links given in the section would be totally incomprehensible for someone that doesn't already understand the topic at a higher level. —  Fly by Night  (  talk  )  19:29, 9 September 2016 (UTC)

 * I think the link to Diophantine_equation is quite helpful. That section of that article gives a precise description of the algorithm, and, despite your objection, I don't think it would be "totally incomprehensible".  True, it doesn't provide an example, and neither does the new section of this article; but it's a good exercise to apply the general algorithm to a special case.  If you think this section of the present article would benefit from continuing the running example, I suggest you add it. Ozob (talk) 18:49, 10 September 2016 (UTC)

 * It is not incomprehensible to someone that already understands the topic - that was my point. But for a casual reader or the majority first year undergraduates, the links are incomprehensible. Heck, I understand the topic and yet find the link baffling. Remember that not everyone is blessed with tremendous mathematical intelligence. With regards to adding an example: I tried that once before and got my fingers burned. Id rather not spend time adding something that will get reverted. — Fly by Night  (  talk  )  00:22, 11 September 2016 (UTC)

Is the coprime condition necessary and sufficient for the ring isomorphism?
In the section 'Theorem statement', the article states "if the $$n_i$$ are pairwise coprime" then we have the ring isomorphism $$\mathbb{Z}/N\mathbb{Z} \cong \mathbb{Z}/n_1\mathbb{Z} \times \cdots \times \mathbb{Z}/n_k\mathbb{Z}$$. I know from my group theory course that the $$n_i$$ are coprime *if and only if* we have the *group* isomorphism. If this is also true for the ring isomorphism then I suggest it be added to the article. Similarly in the section 'Generalization to arbitrary rings', the article states "If the ideals are pairwise coprime, we have the isomorphism", and if this is necessary and sufficient as well then it should be included in the article. — Preceding unsigned comment added by Joel Brennan (talk • contribs) 15:57, 25 April 2019 (UTC)

Uniqueness
CLAIM: Suppose that x and y are both solutions to all the congruences. As x and y give the same remainder, when divided by ni, their difference x − y is a multiple of each ni. As the ni are pairwise coprime, their product N also divides x − y

given N > 1 given x,y < N then -N < x-y < N

if x>y, 0 < x-y < N and 0 < (x-y) / N < 1 so N does not "divide" (x-y)

if x<y, -N < x-y < 0 and -1 < (x-y) / N < 0 so N does not "divide" (x-y)

if x=y, x-y = 0 and (x-y) / N = 0 so only in this case does N does "divide" (x-y)

it is better to simply say x is equivalent to y modulo N, x ≡ y (mod N) — Preceding unsigned comment added by Nonki72 (talk • contribs) 21:32, 19 April 2020 (UTC)

 * The first part of the uniqueness proof is to show that $x ≡ y (mod N)$. This part of the proof does not require that $x, y < N$ as you have assumed. The second part is to show that there is only one solution that is non-negative and less than $N$. It is this that you have provided a correct proof for. The article just indicates that this is so without providing the details. The argument is simple and so there is justification for omitting it. As Wikipedia is an encyclopedia and not a math text (see WP:NOTTEXTBOOK) most proof details are not included.--Bill Cherowitzo (talk) 22:45, 19 April 2020 (UTC)

Inconsistent scope of generalizations
In the introductory text, it says that CRT has been generalized to commutative rings with a formulation involving ideals. After reading this sentence, I thought there will be an entry about generalization to arbitrary commutative rings. But instead, there is only one entry about generalization to arbitrary (not necessarily commutative) rings. — Preceding unsigned comment added by Vincent B. Hwang (talk • contribs) 05:37, 14 December 2021 (UTC)
 * However, the commutative case is much more used that the non-commutative one, and this may explain this discrepancy. D.Lazard (talk) 11:26, 14 December 2021 (UTC)

title: Talk:Chinese remainder theorem


text:
Coprophagia

Coprophagia or coprophagy  is the consumption of feces. The word is derived from the Ancient Greek κόπρος "feces" and φαγεῖν  "to eat". Coprophagy refers to many kinds of feces-eating, including eating feces of other species (heterospecifics), of other individuals (allocoprophagy), or one's own (autocoprophagy) – those once deposited or taken directly from the anus.

In humans, coprophagia has been described since the late 19th century in individuals with mental illnesses and in some sexual acts, such as the practices of anilingus and felching where sex partners insert their tongue into each other's anus and ingest biologically significant amounts of feces. Some animal species eat feces as a normal behavior, in particular lagomorphs, which do so to allow tough plant materials to be digested more thoroughly by passing twice through the digestive tract. Other species may eat feces under certain conditions.

In cuisine
The feces of the rock ptarmigan is used in Urumiit, which is a delicacy in some Inuit cuisine. Several beverages are made using the feces of animals, including but not limited to Kopi luwak, insect tea, and Black Ivory Coffee. Casu martzu is a cheese that uses the digestive processes of live maggots to help ferment and break down the cheese's fats.

As a medical treatment for CDI and other conditions
In Fecal microbiota transplant (FMT), also known as a stool transplant, fecal bacteria and other microbes from a healthy individual are transferred into a patient as an effective treatment for Clostridioides difficile infection (CDI). This treatment has also been used to try to cure other conditions with various results. See: Fecal microbiota transplant.

As a supposed medical treatment
Ayurveda and Siddha medicine use various animal excreta in various forms. The dung and urine of the Zebu is especially important in the list.

Centuries ago (mid 16th century) physicians tasted their patients' feces, to better judge their state and condition, according to François Rabelais, who studied medicine but was also a writer of satirical and grotesque fiction. Further information is needed to confirm the accuracy and context of statement.

Lewin reported, "... consumption of fresh, warm camel feces has been recommended by Bedouins as a remedy for bacterial dysentery; its efficacy (probably attributable to the antibiotic subtilisin from Bacillus subtilis) was anecdotally confirmed by German soldiers in Africa during World War II". However, this story is likely a myth, independent research was not able to verify any of these claims.

As a cult practice
Members of a religious cult in Thailand routinely ate the feces and dead skin of their leader, whom they considered to be a holy man with healing powers.

As a paraphilia
Coprophilia is a paraphilia (DSM-5), where the object of sexual interest is feces, and may be associated with coprophagia. Coprophagia is sometimes depicted in pornography, usually under the term "scat" (from scatology). A notorious example of this is the pornographic shock video 2 Girls 1 Cup. The 120 Days of Sodom, a 1785 novel by Marquis de Sade, is full of detailed descriptions of erotic sadomasochistic coprophagia. The film of the same name also contains scenes of coprophilia and coprophagia.

Coprophagia has also been observed in some people with schizophrenia and pica.

By invertebrates
Coprophagous insects consume and redigest the feces of large animals. These feces contain substantial amounts of semidigested food, particularly in the case of herbivores, owing to the inefficiency of the large animals' digestive systems. Thousands of species of coprophagous insects are known, especially among the orders Diptera and Coleoptera. Examples of such flies are Scathophaga stercoraria and Sepsis cynipsea, dung flies commonly found in Europe around cattle droppings.

Among beetles, dung beetles are a diverse lineage, many of which feed on the microorganism-rich liquid component of mammals' dung, and lay their eggs in balls composed mainly of the remaining fibrous material. Group living and aggregation among common earwigs promotes allo-coprophagy (consuming the feces of other members of one's own species) to promote the growth of helpful gut bacteria and provide a food source when food is scarce.

Through proctodeal feeding, termites eat one another's feces as a means of obtaining their hindgut protists. Termites and protists have a symbiotic relationship (e.g. with the protozoan that allows the termites to digest the cellulose in their diet). For example, in one group of termites, a three-way symbiotic relationship exists; termites of the family Rhinotermitidae, cellulolytic protists of the genus Pseudotrichonympha in the guts of these termites, and intracellular bacterial symbionts of the protists.

By vertebrates
Lagomorphs (rabbits, hares, pikas) and some other mammals ferment fiber in their cecums, which is then expelled as cecotropes and eaten from the anus, a process called "cecotrophy". Then their food is processed through the gastrointestinal tract a second time, which allows them to absorb more nutrition. While cecotropes are expelled from the anus, they are not feces and thus eating them is not called coprophagia.

Domesticated and wild mammals are sometimes coprophagic.

Some dogs may lack critical digestive enzymes when they are only eating processed dried foods, so they gain these from consuming fecal matter. They only consume fecal matter that is less than two days old which supports this theory.

Cattle in the United States are often fed chicken litter. Concerns have arisen that the practice of feeding chicken litter to cattle could lead to bovine spongiform encephalopathy (mad-cow disease) because of the crushed bone meal in chicken feed. The U.S. Food and Drug Administration regulates this practice by attempting to prevent the introduction of any part of cattle brain or spinal cord into livestock feed. Chickens also eat their own feces. Other countries, such as Canada, have banned chicken litter for use as a livestock feed.

The young of elephants, giant pandas, koalas, and hippos eat the feces of their mothers or other animals in the herd, to obtain the bacteria required to properly digest vegetation found in their ecosystems. When such animals are born, their intestines are sterile and do not contain these bacteria. Without doing this, they would be unable to obtain any nutritional value from plants. Piglets with access to maternal feces early in life exhibited better performance.

Hamsters, guinea pigs, chinchillas, hedgehogs, and pigs eat their own droppings, which are thought to be a source of vitamins B and K, produced by gut bacteria. Sometimes, there is also the aspect of self-anointment while these creatures eat their droppings. On rare occasions gorillas have been observed consuming their feces, possibly out of boredom, a desire for warm food, or to reingest seeds contained in the feces.

Coprophagia by plants
Some carnivorous plants, such as pitcher plants of the genus Nepenthes, obtain nourishment from the feces of commensal animals. Notable examples include Nepenthes jamban, whose specific name is the Indonesian word for toilet. Manure is organic matter, mostly animal feces, that is used as organic fertilizer for plants in agriculture.

title: Coprophagia


text:
C. L. Moore

Catherine Lucille Moore (January 24, 1911 – April 4, 1987) was an American science fiction and fantasy writer, who first came to prominence in the 1930s writing as C. L. Moore. She was among the first women to write in the science fiction and fantasy genres (though earlier woman writers in these genres include Clare Winger Harris, Greye La Spina, and Francis Stevens, among others). Moore's work paved the way for many other female speculative fiction writers.

Moore married her first husband Henry Kuttner in 1940, and most of her work from 1940 to 1958 (Kuttner's death) was written by the couple collaboratively. They were prolific co-authors under their own names, although more often under any one of several pseudonyms.

As "Catherine Kuttner", she had a brief career as a  television scriptwriter from 1958 to 1962. She retired from writing in 1963.

Early life
Moore was born on January 24, 1911, in Indianapolis, Indiana. She was chronically ill as a child and spent much of her time reading literature of the fantastic. She left college during the Great Depression to work as a secretary at the Fletcher Trust Company in Indianapolis.

Early career
The Vagabond, a student-run magazine at Indiana University, published three of her stories when she was a student there. The three short stories, all with a fantasy theme and all credited to "Catherine Moore", appeared in 1930/31. Her first professional sales appeared in pulp magazines beginning in 1933. Her decision to publish under the name "C. L. Moore" stemmed not from a desire to hide her gender, but to keep her employers at Fletcher Trust from knowing that she was working as a writer on the side.

Her early work included two significant series in Weird Tales, then edited by Farnsworth Wright. One features the rogue and adventurer Northwest Smith wandering through the Solar System; the other features the swordswoman/warrior Jirel of Joiry, one of the first female protagonists in sword-and-sorcery fiction. Both series are sometimes named for their lead characters. One of the Northwest Smith stories, "Nymph of Darkness" (Fantasy Magazine (April 1935); expurgated version, Weird Tales (Dec 1939)) was written in collaboration with Forrest J Ackerman.

The most famous Northwest Smith story is "Shambleau", which was also Moore's first professional sale. It originally appeared in the November 1933 issue of Weird Tales, netting her $100, and later becoming a popular anthology reprint. Her most famous Jirel story is also the first one, "Black God's Kiss", which was the cover story in the October 1934 issue of Weird Tales, subtitled "the weirdest story ever told" (see figure). Moore's early stories were notable for their emphasis on the senses and emotions, which was unusual in genre fiction at the time.

Moore's work also appeared in Astounding Science Fiction magazine throughout the 1940s. Several stories written for that magazine were later collected in her first published book, Judgment Night (1952) One of them, the novella "No Woman Born" (1944), was to be included in more than 10 different science fiction anthologies including The Best of C. L. Moore.

Included in that collection were "Judgment Night" (first published in August and September 1943), the lush rendering of a future galactic empire with a sober meditation on the nature of power and its inevitable loss; "The Code" (July 1945), an homage to the classic Faust with modern theories and Lovecraftian dread; "Promised Land" (February 1950) and "Heir Apparent" (July 1950), both documenting the grim twisting that mankind must undergo in order to spread into the Solar System; and "Paradise Street" (September 1950), a futuristic take on the Old West conflict between lone hunter and wilderness-taming settlers.

Marriage to Henry Kuttner and literary collaborations
Moore met Henry Kuttner, also a science fiction writer, in 1936 when he wrote her a fan letter under the impression that "C. L. Moore" was a man. They soon collaborated on a story that combined Moore's signature characters, Northwest Smith and Jirel of Joiry: "Quest of the Starstone" (1937).

Moore and Kuttner married in 1940 and thereafter wrote many of their stories in collaboration, sometimes under their own names, but more often using the joint pseudonyms C. H. Liddell, Lawrence O'Donnell, or Lewis Padgett — most commonly the latter, a combination of their mothers' maiden names. Moore still occasionally wrote solo work during this period, including the frequently anthologized "No Woman Born" (1944). A selection of Moore's solo short fiction work from 1942 through 1950 was collected in 1952's Judgement Night. Moore's only solo novel, Doomsday Morning, appeared in 1957.

The vast majority of Moore's work in the period, though, was written as part of a very prolific partnership. Working together, the couple managed to combine Moore's style with Kuttner's more cerebral storytelling. They continued to work in science fiction and fantasy, and their works include two frequently anthologized sci-fi classics: "Mimsy Were the Borogoves" (February 1943), the basis for the film The Last Mimzy (2007), and Vintage Season (September 1946), the basis for the film Timescape (1992). As "Lewis Padgett" they also penned two mystery novels: The Brass Ring (1946) and The Day He Died (1947).

Later career
After Kuttner's death in 1958, Moore continued teaching her writing course at the University of Southern California, but permanently retired from writing any further literary fiction. Instead, working as "Catherine Kuttner", she carved out a short-lived career as a scriptwriter for Warner Bros. television, writing episodes of the westerns Sugarfoot, Maverick, and The Alaskans, as well as the detective series 77 Sunset Strip, all between 1958 and 1962. However, upon marrying Thomas Reggie (who was not a writer) in 1963, she ceased writing entirely.

Moore was the author guest of honor at Kansas City, Missouri's fantasy and science fiction convention BYOB-Con 6, held over the U.S. Memorial Day weekend in May 1976. She was a pro guest of honor at Denvention II (the 39th World Science Fiction Convention) in 1981.

In a 1979 interview, she said that she and a writer friend were collaborating on a fantasy story, and how it could possibly form the basis of a new series. But nothing was ever published.

In 1981, Moore received two annual awards for her career in fantasy literature: the World Fantasy Award for Life Achievement, chosen by a panel of judges at the World Fantasy Convention, and the Gandalf Grand Master Award, chosen by vote of participants in the World Science Fiction Convention. (Thus she became the eighth and final Grand Master of Fantasy, sponsored by the Swordsmen and Sorcerers' Guild of America, in partial analogy to the Grand Master of Science Fiction sponsored by the Science Fiction Writers of America.)

Moore was an active member of the Tom and Terri Pinckard Science Fiction literary salon and a frequent contributor to literary discussions with the regular membership, including Robert Bloch, George Clayton Johnson, Larry Niven, Jerry Pournelle, Norman Spinrad, A. E. van Vogt, and others, as well as many visiting writers and speakers.

Later life
Moore developed Alzheimer's disease, but that was not obvious for several years. She had ceased to attend the meetings when she was nominated to be the first woman Grand Master of the Science Fiction Writers of America; the nomination was withdrawn at the request of her husband, Thomas Reggie, who said the award and ceremony would be at best confusing and likely upsetting to her, given the progress of her disease. She died on April 4, 1987, at her home in Hollywood, California.

Awards

 * 1978: Fritz Leiber Award
 * 1981: Gandalf Grand Master Award, World Fantasy Convention Lifetime Achievement Award
 * 1998: Posthumous induction into the Science Fiction and Fantasy Hall of Fame
 * 2004: Cordwainer Smith Rediscovery Award
 * 2019: Retro Hugo Award for best Novelette for the year 1944

Selected works

 * Earth's Last Citadel (with Henry Kuttner; 1943)
 * The Dark World (credited to Henry Kuttner, but believed by many critics to be a collaboration, 1946)
 * Vintage Season (novella written with Henry Kuttner, as "Lawrence O'Donnell"; 1946). It was filmed in 1992 as Timescape.
 * The Mask of Circe (with Henry Kuttner; 1948, Illustrated by Alicia Austin; 1971)
 * Beyond Earth's Gates (1949)
 * Judgment Night (stories, 1952)
 * Shambleau and Others (stories, 1953)
 * Northwest of Earth (stories, 1954)
 * No Boundaries (with Henry Kuttner; stories, 1955)
 * Doomsday Morning (1957)
 * Jirel of Joiry (Paperback Library, 1969); Black God's Shadow (Donald M. Grant, 1977)—the five Jirel stories collected; the latter a limited edition with color plates, signed, numbered, and boxed
 * The Best of C. L. Moore, edited by Lester Del Rey (Nelson Doubleday, 1975)—includes a biographical introduction by Lester Del Rey, which is carefully noncommittal about the influence of her personal life on her writing, and an autobiographical afterword by Moore
 * Black God's Kiss (Paizo Publishing, 2007; ISBN 978-1-60125-045-2)—the five Jirel stories collected
 * Northwest of Earth: The Complete Northwest Smith (Paizo Publishing, 2008; ISBN 978-1-60125-081-0)—the thirteen Northwest Smith stories collected

title: C. L. Moore


text:
Compactron

Compactrons are a type of vacuum tube, which contain multiple electrode structures packed into a single enclosure. They were designed to compete with early transistor electronics and were used in televisions, radios, and similar roles.

History
The Compactron was a trade name applied to multi-electrode structure tubes specifically constructed on a 12-pin Duodecar base. This vacuum tube family was introduced in 1961 by General Electric in Owensboro, Kentucky to compete with transistorized electronics during the solid state transition. Television sets were a primary application. The idea of multi-electrode tubes itself was far from new and indeed the Loewe company of Germany was producing multi-electrode tubes as far back as 1926, and they even included all of the required passive components as well.

Use was prevalent in televisions because transistors were slow to achieve the high power and frequency capabilities needed particularly in color television sets. The first portable color television, the General Electric Porta-Color, was designed using 13 tubes, 10 of which were Compactrons. Even before the compactron design was unveiled, nearly all tube based electronic equipment used multi-electrode tubes of one type or another. Virtually every AM/FM radio receiver of the 1950s and 60's used a 6AK8 (EABC80) tube (or equivalent) consisting of three diodes and a triode which was designed in 1954.

Compactron's integrated valve design helped lower power consumption and heat generation (they were to tubes what integrated circuits were to transistors). Compactrons were also used in a few high end Hi-Fi stereos. They were also used by Ampeg and Fender in some of their guitar amplifiers. No modern tube based Hi-Fi systems are known to use this tube type, as simpler and more readily available tubes have again filled this niche. One tube, the 7868, is used in some Hi-Fi systems made today. This tube is a Novar tube. It has the same physical dimensions as the compactron, but a 9 pin base. The exhaust tip is on the top or bottom of the tube, depending on the manufacturer's preference. It is currently in production by Electro-Harmonix.(The new power amp, Linear Tube Audio's Ultralinear, uses 4 17JN6 compactron tubes as the power tube in the amp.) The amp generates 20 watts of power with these inexpensive TV tubes.

Notable features
A distinguishing feature of most Compactrons is the placement of the evacuation tip on the bottom end, rather than the top end as was customary with "miniature" tubes, and a characteristic 3/4" diameter circle pin pattern.
 * Most Compactrons ranged in glass envelope diameter from 28 to 70 mm depending upon the internal configuration. Variations of the Compactron design were made by Sylvania and by some Japanese firms.

Examples
Examples of Compactrons type types include:
 * 6AG11 double diode similar to 6AL5, double triode high-mu similar to 12AT7. Designed for FM stereo multiplex service.
 * 6BK11 triple triode. Two of the triodes are similar to 12AX7 and one of them is similar to 5751.
 * 6C10 high-mu triple triode, all three being similar to 12AX7, used for audio amplifiers, and as color matrix amplifiers in television by Sylvania, etc.... not related to the Edison Swan (later Mazda) 6C10 triode-hexode
 * 6M11 twin triode - pentode. Designed for sync separators and AGC amplifier circuits.
 * 6K11 triple triode. Designed for sync separators and AGC amplifier circuits.
 * 6LF6 beam power pentode with anode cap. Designed for horizontal output service.
 * 8B10 twin triode - twin diode. Designed for horizontal phase detector service, and horizontal oscillator service.
 * 12AE10 twin pentode. Designed for FM discriminator/detector, and audio output.
 * 38HK7 pentode diode. Designed for horizontal output service and as a damper diode
 * 1AD2 diode high voltage, used in flyback transformer rectification

Due to their specific applications in television circuits, many different Compactron types were produced. Almost all were assigned using standard US tube numbers.

Technological obsolescence
Integrated circuits (of the analogue and digital type) gradually took over all of the functions that the Compactron was designed for. "Hybrid" television sets produced in the early to mid-1970s made use of a combination of tubes (typically Compactrons), transistors, and integrated circuits in the same set. By the mid-1980s this type of tube was functionally obsolete. Compactrons simply don't exist in any TV sets designed after 1986. Other specialist uses of the tube declined in parallel with the television set manufacture. Manufacture of Compactrons ceased in the early 1990s. New old stock replacements for almost all Compactron types produced are easily found for sale on the Internet.

title: Compactron


text:
Carmichael number

In number theory, a Carmichael number is a composite number $n$ which in modular arithmetic satisfies the congruence relation:
 * $$b^n\equiv b\pmod{n}$$

for all integers $b$. The relation may also be expressed in the form:
 * $$b^{n-1}\equiv 1\pmod{n}$$

for all integers $$b$$ that are relatively prime to $n$. They are infinite in number.

They constitute the comparatively rare instances where the strict converse of Fermat's Little Theorem does not hold. This fact precludes the use of that theorem as an absolute test of primality.

The Carmichael numbers form the subset K1 of the Knödel numbers.

The Carmichael numbers were named after the American mathematician Robert Carmichael by Nicolaas Beeger, in 1950. Øystein Ore had referred to them in 1948 as numbers with the "Fermat property", or "F numbers" for short.

Overview
Fermat's little theorem states that if $$p$$ is a prime number, then for any integer $b$, the number $$b^p-b$$ is an integer multiple of $p$. Carmichael numbers are composite numbers which have the same property. Carmichael numbers are also called Fermat pseudoprimes or absolute Fermat pseudoprimes. A Carmichael number will pass a Fermat primality test to every base $$b$$ relatively prime to the number, even though it is not actually prime. This makes tests based on Fermat's Little Theorem less effective than strong probable prime tests such as the Baillie–PSW primality test and the Miller–Rabin primality test.

However, no Carmichael number is either an Euler–Jacobi pseudoprime or a strong pseudoprime to every base relatively prime to it so, in theory, either an Euler or a strong probable prime test could prove that a Carmichael number is, in fact, composite.

Arnault gives a 397-digit Carmichael number $$N$$ that is a strong pseudoprime to all prime bases less than 307:
 * $$N = p \cdot (313(p - 1) + 1) \cdot (353(p - 1) + 1 )$$

where
 * $$p = $$29674495668685510550154174642905332730771991799853043350995075531276838753171770199594238596428121188033664754218345562493168782883

is a 131-digit prime. $$p$$ is the smallest prime factor of $N$, so this Carmichael number is also a (not necessarily strong) pseudoprime to all bases less than $p$.

As numbers become larger, Carmichael numbers become increasingly rare. For example, there are 20,138,200 Carmichael numbers between 1 and 1021 (approximately one in 50 trillion (5·1013) numbers).

Korselt's criterion
An alternative and equivalent definition of Carmichael numbers is given by Korselt's criterion.

 * Theorem (A. Korselt 1899): A positive composite integer $$n$$ is a Carmichael number if and only if $$n$$ is square-free, and for all prime divisors $$p$$ of $n$, it is true that $p - 1 \mid n - 1$.

It follows from this theorem that all Carmichael numbers are odd, since any even composite number that is square-free (and hence has only one prime factor of two) will have at least one odd prime factor, and thus $$p-1 \mid n-1$$ results in an even dividing an odd, a contradiction. (The oddness of Carmichael numbers also follows from the fact that $$-1$$ is a Fermat witness for any even composite number.) From the criterion it also follows that Carmichael numbers are cyclic. Additionally, it follows that there are no Carmichael numbers with exactly two prime divisors.

Discovery
The first seven Carmichael numbers, from 561 to 8911, were all found by the Czech mathematician Václav Šimerka in 1885 (thus preceding not just Carmichael but also Korselt, although Šimerka did not find anything like Korselt's criterion). His work, published in Czech scientific journal Časopis pro pěstování matematiky a fysiky, however, remained unnoticed.

Korselt was the first who observed the basic properties of Carmichael numbers, but he did not give any examples.

That 561 is a Carmichael number can be seen with Korselt's criterion. Indeed, $$561 = 3 \cdot 11 \cdot 17$$ is square-free and $p_j - 1 \mid n - 1$, $$10 \mid 560$$ and $p_i \mid n - 1$. The next six Carmichael numbers are :
 * $$1105 = 5 \cdot 13 \cdot 17 \qquad (4 \mid 1104;\quad 12 \mid 1104;\quad 16 \mid 1104)$$
 * $$1729 = 7 \cdot 13 \cdot 19 \qquad (6 \mid 1728;\quad 12 \mid 1728;\quad 18 \mid 1728)$$
 * $$2465 = 5 \cdot 17 \cdot 29 \qquad (4 \mid 2464;\quad 16 \mid 2464;\quad 28 \mid 2464)$$
 * $$2821 = 7 \cdot 13 \cdot 31 \qquad (6 \mid 2820;\quad 12 \mid 2820;\quad 30 \mid 2820)$$
 * $$6601 = 7 \cdot 23 \cdot 41 \qquad (6 \mid 6600;\quad 22 \mid 6600;\quad 40 \mid 6600)$$
 * $$8911 = 7 \cdot 19 \cdot 67 \qquad (6 \mid 8910;\quad 18 \mid 8910;\quad 66 \mid 8910).$$

In 1910, Carmichael himself also published the smallest such number, 561, and the numbers were later named after him.

Jack Chernick proved a theorem in 1939 which can be used to construct a subset of Carmichael numbers. The number $$(6k + 1)(12k + 1)(18k + 1)$$ is a Carmichael number if its three factors are all prime. Whether this formula produces an infinite quantity of Carmichael numbers is an open question (though it is implied by Dickson's conjecture).

Paul Erdős heuristically argued there should be infinitely many Carmichael numbers. In 1994 W. R. (Red) Alford, Andrew Granville and Carl Pomerance used a bound on Olson's constant to show that there really do exist infinitely many Carmichael numbers. Specifically, they showed that for sufficiently large $$n$$, there are at least $$n^{2/7}$$ Carmichael numbers between 1 and $n$.

Thomas Wright proved that if $$a$$ and $$m$$ are relatively prime, then there are infinitely many Carmichael numbers in the arithmetic progression $2 \mid 560$, where $16 \mid 560$.

Löh and Niebuhr in 1992 found some very large Carmichael numbers, including one with 1,101,518 factors and over 16 million digits. This has been improved to 10,333,229,505 prime factors and 295,486,761,787 digits, so the largest known Carmichael number is much greater than the largest known prime.

Factorizations
Carmichael numbers have at least three positive prime factors. The first Carmichael numbers with $$k = 3, 4, 5, \ldots$$ prime factors are :

The first Carmichael numbers with 4 prime factors are :

The second Carmichael number (1105) can be expressed as the sum of two squares in more ways than any smaller number. The third Carmichael number (1729) is the Hardy-Ramanujan Number: the smallest number that can be expressed as the sum of two cubes (of positive numbers) in two different ways.

Distribution
Let $$C(X)$$ denote the number of Carmichael numbers less than or equal to $n$. The distribution of Carmichael numbers by powers of 10 :

In 1953, Knödel proved the upper bound:
 * $$C(X) < X \exp\left({-k_1 \left( \log X \log \log X\right)^\frac{1}{2}}\right)$$

for some constant $a + k \cdot m$.

In 1956, Erdős improved the bound to
 * $$C(X) < X \exp\left(\frac{-k_2 \log X \log \log \log X}{\log \log X}\right)$$

for some constant $k = 1, 2, \ldots$. He further gave a heuristic argument suggesting that this upper bound should be close to the true growth rate of $X$.

In the other direction, Alford, Granville and Pomerance proved in 1994 that for sufficiently large X,
 * $$C(X) > X^\frac{2}{7}.$$

In 2005, this bound was further improved by Harman to
 * $$C(X) > X^{0.332}$$

who subsequently improved the exponent to $k_1$.

Regarding the asymptotic distribution of Carmichael numbers, there have been several conjectures. In 1956, Erdős conjectured that there were $$X^{1-o(1)}$$ Carmichael numbers for X sufficiently large. In 1981, Pomerance sharpened Erdős' heuristic arguments to conjecture that there are at least
 * $$ X \cdot L(X)^{-1 + o(1)} $$

Carmichael numbers up to $k_2$, where $C(X)$.

However, inside current computational ranges (such as the counts of Carmichael numbers performed by Pinch up to 1021), these conjectures are not yet borne out by the data.

In 2021, Daniel Larsen proved an analogue of Bertrand's postulate for Carmichael numbers first conjectured by Alford, Granville, and Pomerance in 1994. Using techniques developed by Yitang Zhang and James Maynard to establish results concerning small gaps between primes, his work yielded the much stronger statement that, for any $$\delta>0$$ and sufficiently large $$x$$ in terms of $$\delta$$, there will always be at least
 * $$\exp{\left(\frac{\log{x}}{(\log \log{x})^{2+\delta}}\right)} $$

Carmichael numbers between $$x$$ and
 * $$x+\frac{x}{(\log{x})^{\frac{1}{2+\delta}}}.$$

Generalizations
The notion of Carmichael number generalizes to a Carmichael ideal in any number field $0.7039 \cdot 0.4736 = 0.33336704 > 1/3$. For any nonzero prime ideal $$\mathfrak p$$ in $X$, we have $$\alpha^{{\rm N}(\mathfrak p)} \equiv \alpha \bmod {\mathfrak p}$$ for all $$\alpha$$ in $L(x) = \exp{ \left( \frac{\log x \log\log\log x} {\log\log x} \right) }$, where $${\rm N}(\mathfrak p)$$ is the norm of the ideal $K$. (This generalizes Fermat's little theorem, that $$m^p \equiv m \bmod p$$ for all integers ${\mathcal O}_K$ when ${\mathcal O}_K$ is prime.) Call a nonzero ideal $$\mathfrak a$$ in $${\mathcal O}_K$$ Carmichael if it is not a prime ideal and $$\alpha^{{\rm N}(\mathfrak a)} \equiv \alpha \bmod {\mathfrak a}$$ for all $\mathfrak p$, where $${\rm N}(\mathfrak a)$$ is the norm of the ideal $m$. When $p$ is $\alpha \in {\mathcal O}_K$, the ideal $$\mathfrak a$$ is principal, and if we let $\mathfrak a$ be its positive generator then the ideal $$\mathfrak a = (a)$$ is Carmichael exactly when $K$ is a Carmichael number in the usual sense.

When $\mathbf Q$ is larger than the rationals it is easy to write down Carmichael ideals in $a$: for any prime number $a$ that splits completely in $K$, the principal ideal $$p{\mathcal O}_K$$ is a Carmichael ideal. Since infinitely many prime numbers split completely in any number field, there are infinitely many Carmichael ideals in ${\mathcal O}_K$. For example, if $p$ is any prime number that is 1 mod 4, the ideal $K$ in the Gaussian integers $$\mathbb Z[i]$$ is a Carmichael ideal.

Both prime and Carmichael numbers satisfy the following equality:
 * $$\gcd \left(\sum_{x=1}^{n-1} x^{n-1}, n\right) = 1.$$

Lucas–Carmichael number
A positive composite integer $$n$$ is a Lucas–Carmichael number if and only if $$n$$ is square-free, and for all prime divisors $$p$$ of ${\mathcal O}_K$, it is true that $p$. The first Lucas–Carmichael numbers are:
 * 399, 935, 2015, 2915, 4991, 5719, 7055, 8855, 12719, 18095, 20705, 20999, 22847, 29315, 31535, 46079, 51359, 60059, 63503, 67199, 73535, 76751, 80189, 81719, 88559, 90287, ...

Quasi–Carmichael number
Quasi–Carmichael numbers are squarefree composite numbers $(p)$ with the property that for every prime factor $n$ of $p + 1 \mid n + 1$, $n$ divides $p$ positively with $n$ being any integer besides 0. If $p + b$, these are Carmichael numbers, and if $n + b$, these are Lucas–Carmichael numbers. The first Quasi–Carmichael numbers are:
 * 35, 77, 143, 165, 187, 209, 221, 231, 247, 273, 299, 323, 357, 391, 399, 437, 493, 527, 561, 589, 598, 713, 715, 899, 935, 943, 989, 1015, 1073, 1105, 1147, 1189, 1247, 1271, 1295, 1333, 1517, 1537, 1547, 1591, 1595, 1705, 1729, ...

Knödel number
An n-Knödel number for a given positive integer n is a composite number m with the property that each $b$ coprime to m satisfies $b = -1$. The $b = 1$ case are Carmichael numbers.

Higher-order Carmichael numbers
Carmichael numbers can be generalized using concepts of abstract algebra.

The above definition states that a composite integer n is Carmichael precisely when the nth-power-raising function pn from the ring Zn of integers modulo n to itself is the identity function. The identity is the only Zn-algebra endomorphism on Zn so we can restate the definition as asking that pn be an algebra endomorphism of Zn. As above, pn satisfies the same property whenever n is prime.

The nth-power-raising function pn is also defined on any Zn-algebra A. A theorem states that n is prime if and only if all such functions pn are algebra endomorphisms.

In-between these two conditions lies the definition of Carmichael number of order m for any positive integer m as any composite number n such that pn is an endomorphism on every Zn-algebra that can be generated as Zn-module by m elements. Carmichael numbers of order 1 are just the ordinary Carmichael numbers.

An order-2 Carmichael number
According to Howe, 17 · 31 · 41 · 43 · 89 · 97 · 167 · 331 is an order 2 Carmichael number. This product is equal to 443,372,888,629,441.

Properties
Korselt's criterion can be generalized to higher-order Carmichael numbers, as shown by Howe.

A heuristic argument, given in the same paper, appears to suggest that there are infinitely many Carmichael numbers of order m, for any m. However, not a single Carmichael number of order 3 or above is known.

title: Carmichael number