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Muggles' Guide to Harry Potter/Characters/Remus Lupin

Overview
Remus John Lupin is a wizard who joins the cast of characters in Harry Potter and the Prisoner of Azkaban as the new Defence Against the Dark Arts teacher.

Lupin's birthday is March 10th, according to the author.

Etymology
The etymology of the character's name is: This is an example of the author's use of perhaps excessively appropriate nomenclature. The events which made this name so correct for Lupin must have occurred at some time after he was named. While this might raise a few eyebrows in mature readers, we remind students that this is primarily a book series for children, and as such having names that directly link to characteristics of the named actors is appropriate.
 * Remus: In Roman legend Romulus and Remus were the founders of Rome who were raised by a she-wolf.
 * Lupin: Derived from Latin "lupinus" of a wolf.

Prisoner of Azkaban
Remus J. Lupin is introduced on the Hogwarts Express, where he spends almost the entire trip sleeping. Even asleep, however, he assists Harry and his friends; his presence prevents Draco Malfoy from engaging in any mischief. When a Dementor hunting the fugitive Sirius Black, enters the compartment, Lupin awakes and repels it with a Charm. He then provides chocolate to the frightened students as a restorative and sends an owl ahead to inform the school nurse that Harry may need medical attention. Madam Pomfrey later commends his actions, saying that at last they have a Defence Against the Dark Arts teacher who knows medical remedies.

At the Welcoming Feast, Dumbledore announces that Professor Lupin is the new Defence Against the Dark Arts teacher — something Harry, Ron, and Hermione already guessed. In his first-ever lesson with Harry's class, Lupin demonstrates an unexpected ability in subduing Peeves. He also gives Neville Longbottom a much-needed boost in confidence by having him lead an exercise to ward off a Boggart. Lupin's practical, hands-on teaching methods are popular with students, making him an instant favourite.

When Harry later asks why Lupin prevented him from tackling his own Boggart, Lupin says he feared it would turn into Lord Voldemort and frighten the class. Lupin is impressed that Harry's worst fear is not Voldemort, but Dementors. Shortly after, Severus Snape, brings Lupin a potion and leaves. Harry suggests that Snape, who has long coveted the Defence Against the Dark Arts job and apparently despises Lupin, may have poisoned the potion, but Lupin drinks it without hesitation.

After Dementors cause Harry to faint and fall off his broom during a Quidditch match, he asks Lupin to teach him how to repel them. Lupin shows Harry the Patronus charm. During these lessons, Harry learns that Lupin, Sirius Black, and Harry's father were schoolmates. During the next Quidditch match, Harry successfully conjures the charm to repel the Dementors, although it is actually Malfoy and his cronies in disguise. Lupin congratulates Harry on his Patronus, but he is rather shaken by the shape it took.

Lupin plays a major role in the Shrieking Shack scene. Upon entering the shack, he immediately disarms Harry, Ron, and Hermione, who have overpowered Sirius Black. Certain that Black is innocent, Lupin returns the Trio's wands and holsters his own as a reassuring gesture that he trusts everyone. That's where Hermione reveals to everyone that Lupin is a werewolf. However, he truthfully says he was not after harming anyone, and after a while, they believe him but not Sirius. Lupin defends his friend and tells them about the friendship between himself, James Potter, Sirius Black, and Peter Pettigrew, the four Marauders who created the Marauder's Map. Severus Snape bursts in, accusing Lupin of abetting Black, but the Trio knock him unconscious with a combined charm. Both Lupin and Black then unmask Scabbers, Ron's rat, as Peter Pettigrew in his Animagus form. James was also an Animagus, as is Sirius. It was Pettigrew who betrayed James and Lily to Voldemort. As they prepare to execute Pettigrew, Harry intervenes, saying that his father would not want his two best friends to become murderers; instead, the Dementors can have him. While heading to the castle, the full moon rises. Lupin, who has forgotten his potion, transforms into a dangerous werewolf. Black, in his Animagus dog form, is unable to subdue Lupin but chases him away into the Forbidden Forest. Black is captured by Snape and imprisoned in the castle; Harry and Hermione rescue him, and he escapes on Buckbeak, a Hippogriff wrongly sentenced to death.

Denied the fame that Black's capture would have bestowed upon him, Snape takes petty revenge by revealing to the entire school that Lupin is a werewolf. Lupin resigns his post, knowing parents will object to having such a person teaching their children. As Lupin packs to leave, he tells Harry that his father's Animagus form was a stag, the same as Harry's Patronus. That is why Lupin was so affected when he saw it at the Quidditch match.

Goblet of Fire
Lupin does not directly appear in the fourth book. After Voldemort's return to power and Cornelius Fudge's departure, Dumbledore tells Sirius Black to alert "the old crowd," naming Lupin as one of their number, and instructs Sirius to lay low at Remus' until Dumbledore gets in touch with him again.

Order of the Phoenix
Lupin returns in Harry Potter and the Order of the Phoenix when he arrives at the Dursleys to escort Harry to Number Twelve, Grimmauld Place, the Order of the Phoenix headquarters. Lupin stays at Grimmauld Place with Sirius Black, leaving occasionally to carry out unknown tasks for the Order. Harry only sees him at Grimmauld during the school holiday or when he communicates with him and Sirius via the Floo Network. When Harry watches Severus Snape's memories in a Pensieve, he sees his father and Sirius tormenting Snape. Disturbed by James' bullying, he secretly uses the fireplace in Dolores Umbridge's office to contact Sirius and Lupin and asks them about his father's behavior. Lupin says that the young James (and also Sirius) was a bully, but that he eventually outgrew it. His explanation does little to soothe Harry.

Lupin participates in the battle against the Death Eaters in the Department of Mysteries at the book's conclusion. When Harry runs after a stricken Sirius, whose body has floated through a mysterious veiled archway to death, it is Lupin who forcefully restrains him, saving him from the same fate.

Half-Blood Prince
In the sixth book, Lupin acts as a spy for the Order of the Phoenix by infiltrating a dangerous werewolf pack headed by Fenrir Greyback that Voldemort recruited. In the ending, Lupin helps fight the Death Eaters, who have invaded Hogwarts to murder Dumbledore. It is also revealed that Order of the Phoenix member Nymphadora Tonks is in love with Remus. However, Lupin tells Tonks it is too dangerous to be romantically involved with a werewolf, that he is too old for her, and says she deserves someone better. However, Lupin and Tonks are later seen holding hands, which indicates that they have overcome these obstacles and have started a cautious relationship.

Deathly Hallows
Lupin is among the guards escorting Harry and the Harry Potter decoys from the Dursley residence to a safe house. As the escape plan is being outlined in the Dursley kitchen, Tonks is seen wearing a ring; we discover that she and Lupin are married. The Order's plan to move Harry was uncovered, and Lupin arrives at The Burrow with an injured George Weasley in tow. As Mrs. Weasley tends George's wound, Lupin quizzes Harry to confirm his identity. After the other Harry decoys and protectors have arrived, Lupin leaves with Bill Weasley to search for Alastor Moody, who is missing and presumed dead.

Lupin and Tonks are at the Burrow for Harry's birthday, but they quickly depart when Arthur Weasley's Patronus warns them that he is arriving shortly with Rufus Scrimgeour. At Bill and Fleur's wedding, Lupin tells Harry that Werewolves are presently in very poor standing with the Ministry. During the wedding reception, as the Ministry falls and Death Eaters break the existing protective spells surrounding the Burrow, we see Lupin and Tonks, back-to-back, casting protective spells.

A few days after Harry, Hermione, and Ron have settled into Grimmauld Place, Lupin visits. He chastises Ron and Hermione for dropping their guard too quickly when he first identified himself. He knows the Trio has a secret mission, and offers his help. Asked about Tonks, he reports that she is pregnant and is safe at her parents' house. Harry is angered that Lupin wants to go adventuring while Tonks and her unborn child are left at home. Parents, he says, should not abandon their children. Heated words are exchanged, and firing a Jinx that knocks Harry down, Lupin angrily storms out.

In March, Ron tunes in the pirate wireless program "Potterwatch". Lee Jordan, as "River", introduces "Romulus", easily identifiable by his voice as Lupin, who provides an update on Harry. Lupin says that it is certain that Harry is still alive, as his death would be trumpeted far and wide by the regime as a way of eliminating a rallying point for the resistance. Lupin further says that, if Harry is listening, he should follow his instincts, which are nearly always right. Harry and Hermione are both left in tears by this avowal of reconciliation. Lupin goes on to say that Xenophilius Lovegood had been arrested, and that Rubeus Hagrid, who had held a "Support Harry Potter" party in his house, had escaped arrest and was now at large.

When Harry escapes Malfoy Manor and retreats to Shell Cottage, Lupin visits, announcing that Tonks has given birth to a boy they named Teddy. Surprisingly, given their previous acrimonious parting, Lupin is pleased to see Harry and asks him to be Teddy's godfather, which Harry accepts.

Just prior to the final battle with Voldemort at Hogwarts, tension mounts in the Room of Requirement when Ginny Weasley stubbornly insists on joining the fight and by Percy Weasley's unexpected arrival to reconcile with his family. Fleur attempts to ease the situation by asking Lupin about the new baby; Lupin produces a photograph, although no one seems interested in seeing it. Finally, after Percy is forgiven and Ginny is ordered to stay put inside the Room of Requirement, Lupin goes to the Great Hall to command the forces defending the school grounds. Although Tonks arrives, and we see her looking for Lupin as she takes part in the battle, Lupin is not seen again until the battle is over, and his and Tonks' bodies are lying among the dead.

After reliving Snape's memories in the Pensieve, Harry realizes that it was always his fate to face and be killed by Voldemort. Finding his courage unequal to this, and unable to produce a Patronus to pass the Dementors in the Forbidden Forest, Harry summons Lupin's spirit, along with Sirius Black, James Potter, and Lily Potter. Acting as Patronuses, the four spirits shield Harry as he travels through the Forest to Voldemort's encampment. When Harry reaches the Death Eaters, he drops the Resurrection Stone that summoned the shades, causing them to vanish.

Strengths
Remus Lupin is a talented and powerful wizard as well as extremely intelligent. He is thoughtful and tends to think through a situation before acting. He often demonstrates great self-control, remaining calm in stressful situations. Although Lupin has been portrayed as an outcast, he is kind and extraordinarily loyal to his friends and allies, especially Harry.

Lupin was considered by most students and teachers to be the best Defence Against the Dark Arts teacher at Hogwarts during the series. He brought patience, understanding, and humor to the class and greatly encouraged his students, especially Harry and Neville.

Lupin is a true Gryffindor, bravely risking his life to infiltrate Fenrir Greyback's werewolf pack to act as Dumbledore's spy, and loyally serving in the Order of Phoenix. After realizing that Sirius Black was innocent, he immediately went to his aid in the Shrieking Shack where Pettigrew was unmasked as the true traitor.

Weaknesses
At age four, Lupin became afflicted with lycanthropy after being bitten by the werewolf Fenrir Greyback. Every full moon an afflicted person transforms into a wolf-like creature, and, like a rabid dog, is unable to control their actions. Werewolves, transformed, will more or less mindlessly seek out unafflicted humans to bite and infect them. Although there is no cure, a potion has recently been discovered that allows them to retain their sanity during transformations. Lupin, taking this potion, is able to curl up quietly in his office and rest, rather than rampaging through the Forbidden Forest. However, this potion must be taken every day for about a week before a full moon, and it is effective only for one lunar cycle; if the potion is missed, the werewolf again becomes dangerous.

Because they are so dangerous, werewolves are generally feared and shunned by the wizarding community, making them outcasts. Despite this, Albus Dumbledore arranged for the young Remus to attend Hogwarts, where he was confined during full moons. Though Lupin's condition was kept secret, he was always shy and insecure, and made relatively few friends save for Sirius Black, James Potter, and Peter Pettigrew. As a result, the adult Lupin is mostly a loner and has had difficulty finding employment; he always expects to be treated badly.

Relationships with Other Characters
Lupin, a werewolf since he was a young boy, had mostly lived a solitary life with relatively few close friends. As a student at Hogwarts, he was nicknamed "Moony" and was one of the four Marauders, along with his fellow Gryffindor Housemates, James Potter (Prongs), Peter Pettigrew (Wormtail), and Sirius Black (Padfoot). The four became particularly close friends, and though Lupin's condition was kept hidden from students, the other three discovered his secret. They illegally became Animagi, initially to support him, but also to accompany him during his werewolf transformations, releasing him from his monthly confinement in the Shrieking Shack. As the werewolf potion had not been invented yet, Lupin was extremely dangerous when transformed; the other three Marauders, protected while in their animal forms, kept Lupin in check during their nighttime sojourns into the Forbidden Forest and the surrounding countryside. Lily Evans (later Lily Potter) also became Lupin's close friend and confidant while they were at Hogwarts, unconcerned and non-judgmental regarding his condition. Severus Snape, who the other Marauders detested, was also a Hogwarts student at that time. After James and Lily were murdered (as Peter Pettigrew presumably was), and Sirius was sentenced to Azkaban, Lupin was once again mostly friendless.

Lupin greatly admired Professor Dumbledore, and was always grateful for his allowing him to attend Hogwarts, despite his being a werewolf. Dumbledore continued to help Lupin by hiring him as the Defense Against the Dark Arts teacher in book 3, knowing Lupin would be a highly competent instructor. Lupin was an immediate favourite, except with Slytherin students, who continually mocked his shabby robes. Not only the students liked him; he was also well-regarded by the faculty, due to his congenial personality and evident competence. Only Snape disliked him, still holding a grudge against Lupin for his close association with his two Hogwarts nemeses: James Potter and Sirius Black. Snape also coveted the DADA position that Lupin occupied. Despite Snape's animosity, he diligently prepared Lupin's monthly werewolf potion, though, angry when Lupin aided fugitive Sirius Black's escape, later informed students about Lupin's condition, forcing Lupin to resign.

Lupin also joined the Order of the Phoenix, and grew close to Arthur and Molly Weasley, also members. He was on friendly terms with "Mad Eye" Moody, another key Order of the Phoenix member who appeared to regard Lupin favourably. Sirius Black and Lupin were reunited after Sirius escapes Azkaban prison, though he was killed two years later during the Battle at the Ministry. In later books, Lupin became a reassuring, stable presence to Harry. While Harry's godfather, Sirius Black, had more claim to Harry's affection, Harry was occasionally alarmed by Sirius' impetuous and sometimes erratic behavior. Harry would have liked having more time to talk with Lupin, who by then was often away on Order of the Phoenix business, giving Harry less opportunity to spend time with him.

Lupin was close to Harry, and became more so after Sirius' death, though never in the same way as Harry and Sirius were. Sirius had been Harry's father-figure, while Lupin was like an uncle. Lupin continued to guide Harry, who was still developing as a wizard. In Deathly Hallows, they had a brief falling out, however, when Lupin intended to leave Tonks, believing he was an unfit husband and father, and that she would be better off without him. Harry, having lost so many loved ones, was furious that Lupin would consider abandoning his wife and unborn child during such perilous times. Though Harry's harsh words deeply wounded Lupin, causing him to storm out in anger, he later realized Harry was right, and he returned to Tonks. He and Harry eventually reconcile, and Harry became godfather to Lupin and Tonks' newborn son, Teddy.

The burgeoning relationship between Lupin and Nymphadora Tonks began at the end of Harry Potter and the Half-Blood Prince. Perhaps characteristically, Lupin tried to prevent it, believing he was an unsuitable match and that it was too dangerous a time to consider starting a family. This shows another side to Lupin: he cared so much about the larger picture that he sometimes neglected the smaller one. By refusing to risk bringing a child into the mess that seemed to be their future, he caused pain to himself and to Tonks in their present. He and Tonks overcame these differences and married, having a son (Teddy) soon after.

Professor Lupin
Unlike with some previous Defence Against the Dark Arts instructors, the other professors are pleased with Professor Lupin's appointment – Poppy Pomfrey comments approvingly about him knowing remedies, and in a later book, Professor McGonagall refers to him as a "competent teacher" – high praise indeed from her. The one exception is Severus Snape, who, apart from coveting the Dark Arts position for himself, still harbours deep resentments against Lupin for his association with the other Marauders during their Hogwarts years, though it was James Potter and Sirius Black who tormented Snape.

Lupin immediately wins the entire class' respect, first by putting Peeves to flight, and second, by teaching a truly useful and practical first lesson. His caring about, and warmth towards his students becomes visible by numerous small actions – he calls students by their first names, rather than their surnames as is traditional in the Engish school system – and also by larger things: he has Neville take the first turn at the Boggart, and grants him last shot as well, thereby boosting his confidence. He also freely explains why he prevented Harry from facing the Boggart, and tells Harry that his reaction to the Dementor does not indicate weakness. His volunteering to teach Harry to fight Dementors reflects his nature; he cares, so he will help.

Loyalty to Dumbledore
Growing up as a werewolf made Lupin into a loner and outcast but he was immensely loyal to his friends. After Lupin became a werewolf, his parents believed it impossible for him to attend Hogwarts. However, Dumbledore was sympathetic, and told the Lupins that he would never deny Remus an education as long as certain precautions were implemented. After Lupin left Hogwarts, he found it nearly impossible for a werewolf to find a job in the wizarding community. Dumbledore eventually offered him the position as Defense Against the Dark Arts teacher. Because Dumbledore was so kind to him, Lupin proved to be a devoted ally to the Headmaster when the Order of the Phoenix was re-formed, risking his life to carry out Dumbledore's mission to infiltrate Fenrir Greyback's werewolf pack.

School years
When Lupin's three best friends, James Potter, Sirius Black, and Peter Pettigrew discovered he was a werewolf, they decided to support and protect him during his transformations. Because a werewolf is only dangerous to humans, they secretly learned to become Animagi (transforming into an animal at will). An Animagus must register with the Ministry of Magic, although the three never did. During a full moon, they used their new powers to roam Hogwarts' grounds, and later Hogsmeade village, with Lupin in his wolf-like form. When Lupin was with them, he had more control over his actions during his transformations. He was not entirely harmless, however, and his friends carefully guarded him.

Questions

 * 1) Why are werewolves shunned by people?
 * 2) Lupin mentions that he was a Gryffindor prefect, in part so that he would rein in the other three Marauders. Why didn't Lupin try harder to keep his friends out of trouble when they were at Hogwarts?
 * 3) Hermione identifies Lupin on the Hogwarts Express by the name embossed in peeling letters on his case, "Professor R. J. Lupin". Since one is designated Professor only when starting teaching, and Lupin is freshly appointed, why is the embossing old enough to be damaged already?
 * 4) Why is Remus Lupin so inspirational?

Greater Picture
Harry loses many paternal role models throughout the series. His father, James, was murdered by Voldemort when Harry was only one-year-old. In Harry Potter and the Order of the Phoenix, his godfather, Sirius Black is killed by Bellatrix Lestrange; his body floats through an arched stone portal covered by a fluttering curtain called the Veil – both literally and metaphorically a veil that he passes beyond, never to return. In Harry Potter and the Half-Blood Prince, Dumbledore is killed. In response, Harry feels he can no longer rely on any one person to be his mentor, although, to a lesser extent, Mr. Weasley helps fulfill this role.

It also becomes apparent that Harry has formed a similar attachment to Remus Lupin ever since Harry Potter and the Prisoner of Azkaban. However, as Harry spends little time with Remus in the later books, this relationship progresses little beyond Harry Potter and the Order of the Phoenix. In Harry Potter and the Deathly Hallows, Harry has largely outgrown his need for a father figure, and while he continues to like and respect Lupin, he does not cavil at accusing him of cowardice and abandoning his wife and unborn child. While we are unable to see Lupin's complete reaction to this well-founded accusation, when his son is born, Lupin has not only forgiven Harry, but asks him to be the child's godfather. This leads us to believe that, when given time to reflect on Harry's words, Lupin found them true and accepted their message, no matter that it was delivered hurtfully.


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Muggles' Guide to Harry Potter/Characters/Peter Pettigrew

Overview
Peter Pettigrew, almost always referred to by his nickname "Wormtail", was a friend of James Potter at Hogwarts. Unpopular, meek, and magically mediocre, he ingratiated himself with James, Sirius Black, and Remus Lupin as a means to enhance and protect himself. At Hogwarts, the foursome became known as "The Marauders." He learns, like two of the others, to become an Animagus, in his case taking a rat form. Immediately following Lord Voldemort's disappearance, Sirius Black was convicted (without a trial) and sentenced to life imprisonment in Azkaban prison for Pettigrew's supposed murder.

Pettigrew's wand is chestnut and dragon heartstring.

Philosopher's Stone
While he appears in this book, his identity has not yet been revealed. He is merely known as "Scabbers", Ron's pet rat.

Chamber of Secrets
While he appears in this book, his identity still remains unknown.

Prisoner of Azkaban
At the throughout the story, "Scabbers" is revealed to be looking increasingly sickly. Hermione Granger's Kneazle-cat hybrid Crookshanks is shown repeatedly to try pouncing on him and it is teased that Crookshanks ate him. Peter Pettigrew, on the other hand, is talked about by Professor McGonagall and Cornelius Fudge, overheard by Harry, Ron and Hermione. They refer to him as having been close friends with James Potter and Sirius Black when he was at school, whilst never quite having their talent. It is in this scene that we learn that Peter Pettigrew is supposed to have been dead for many years because Sirius killed him and several Muggles in an explosion. At the conclusion, it is revealed that Ron's pet is actually Pettigrew in his Animagus form and that Crookshanks had seen him for what he was and had been trying to bring him to Sirius. It is further revealed that it was he, Pettigrew, and not Sirius who betrayed the Potters to Lord Voldemort. After Sirius and Remus Lupin force him back into his human form, they attempt to execute him for betraying James and Lily Potter to Lord Voldemort. Harry Potter intervenes and spares his life, believing his father would not want his two best friends to become murderers. As Pettigrew is escorted back to the castle to be handed over to the Dementors, the full moon rises. Lupin transforms into a werewolf; having forgotten to take his potion, he becomes dangerous. Sirius changes into his dog form to protect the others. During the resulting commotion, Pettigrew escapes and sets out to be reunited with his master, Lord Voldemort. Professor Dumbledore later tells Harry that his merciful act created a powerful magical pact; Pettigrew now owes Harry a life debt that may override his loyalty to Voldemort.

Goblet of Fire
In the early parts of the book, Pettigrew cares for Lord Voldemort, who has partially regained his body. We see him in the first chapter, in the Riddle house, where he is heard in conversation with Voldemort, and with the muggle Frank Bryce. There, they are discussing the death of Bertha Jorkins, and plans for the death of Harry Potter, with the assistance of Voldemort's "one true servant" at Hogwarts. Pettigrew is somewhat upset that Voldemort does not seem to see him as a true servant, saying that he has cared for Voldemort in their travels, though Voldemort dismisses this, saying that Pettigrew is only there because he wants to have some of Voldemort's power and has nowhere else to go.

Shortly after Harry finds Mr. Crouch in the Forbidden Forest, Harry has a dream in which he sees Pettigrew being tortured by Voldemort for having messed something up.

At the conclusion, Harry and Cedric Diggory are transported from the Triwizard Third Task maze to a cemetery where the Riddle family is interred. Pettigrew instantly kills Cedric on Voldemort's command, and binds Harry to a tombstone. During a ritual to restore Voldemort to full power, Pettigrew drops Riddle Sr.'s bone, some of Harry's blood, and his own severed hand into a large cauldron—Lord Voldemort emerges fully restored. The Dark Lord uses Pettigrew's Dark Mark to summon the remaining Death Eaters and rewards his servant with a magical silvery hand. Harry barely escapes back to Hogwarts with Cedric's body.

Barty Crouch then reveals to Harry that Pettigrew had brought Bertha Jorkins to Voldemort and had assisted in her torture and murder; through Bertha, they had learned of Barty's still being alive and free, and had traveled to Mr. Crouch's home and placed Bartemius Crouch under the Imperius curse; and it had been Barty and Pettigrew who had captured Mad-Eye Moody. Pettigrew had been responsible for Mr. Crouch gaining his freedom and making his way to Hogwarts; he had been suitably punished for his lapse of attention, and Barty had eliminated the problem by killing Bartemius, his father.

Order of the Phoenix
Pettigrew appears in this book, much younger, in the Pensieve, friends of Sirius Black, Remus Lupin, and James Potter. In this episode, James and Sirius are tormenting a young Severus Snape; neither Lupin nor Pettigrew takes part in this, though they are present.

Half-Blood Prince
Pettigrew is seen only in the first chapter of the book, where he is acting as Severus Snape's servant, and possibly spying on Snape for Voldemort. Snape orders him to bring drinks for his guests; Pettigrew complains that this is not why he was sent to Snape, but Snape overrules him. When Narcissa Malfoy starts outlining why she has come to see Snape, Snape stops her. He fires a spell through the bookcase and apparently catches Pettigrew listening in; Pettigrew is heard to yelp and retreat.

Deathly Hallows
At the beginning of the book, Voldemort is interrupted by loud moaning from under the room in which they are meeting. Voldemort dispatches Pettigrew to quiet their "guest."

Harry is captured by Snatchers, and is brought to Malfoy Manor, along with Ron, Hermione, Dean Thomas, and Griphook. All except Hermione are taken to a downstairs room which is acting as a prison by Pettigrew. There, they find Voldemort's "guest", Mr. Ollivander, and Luna Lovegood. When Dobby appears, Harry has him take Dean, Luna, and Ollivander to safety. Pettigrew is sent to investigate the noise of their departure.

He is attacked by Harry and Ron, who are the only two left in the dungeon. Pettigrew attempts to stop Harry overpowering him by choking him with his silver hand, but is reminded by Harry of the life debt he owes. As a result, he hesitates in killing Harry, showing mercy. In retaliation for this positive action, his silver hand turns on Pettigrew, choking him to death. Harry and Ron are unable to stop it, and leave Pettigrew's dead body behind when they exit the dungeon.

Strengths
Pettigrew was given a silver hand by Lord Voldemort, and this hand demonstrates superhuman strength. It may have certain other special abilities, and possibly even some amount of sentience.

He is also an unregistered Animagus, taking the form of a rat.

While he is said to be a relatively unskilled wizard, he shows occasional moments of brilliance, such as creating a rudimentary body for Lord Voldemort, and eventually assisting in his resurrection. He also never seems to have any problems casting spells or even dueling, suggesting that his perceived lack of skill is either an appearance he chooses to project, or a way for other characters to insult him.

Weaknesses
Peter Pettigrew's loyalty is first and foremost to himself. He is loyal to others only when it serves his own interests and purpose. His friendship to James Potter, Remus Lupin, and Sirius Black was mainly because he was weak, and they protected him. He betrayed them when he decided that his best course lay with Voldemort. In Book 3, he vainly attempts to renew his friendship with Sirius and Lupin when his life is on the line. His appeal to Harry only partially works. Although Harry prevents Black and Lupin from executing him, Harry intends to turn Pettigrew over to the Dementors, making death a more appealing option. While his disguise as a rat is very effective and good for spying or escaping, it also leaves him vulnerable to larger creatures.

Relationships with Other Characters
Pettigrew is a very venal character, largely out for the main chance. He tends to ally himself with whoever he sees as being the strongest at the moment, but his loyalty is entirely to himself; he will be loyal to someone as long as nobody comes along who he thinks will do a better job of protecting him.

Before his identity is revealed, he is in disguise as Scabbers, Ron's rat, and like most rats he remains loyal to Ron only as long as he feels Ron can protect him, and as long as he receives some benefit from the association. Until the third book, Scabbers defends himself and Ron against Ron's attackers, but in the third book Crookshanks seems intent on killing him. With the earlier appearance of Sirius Black, association with Ron becomes untenable, so Pettigrew fakes his death, incriminating Crookshanks, and tries to make his own way. It is telling that when he is rediscovered, Scabbers attempts repeatedly to escape from Ron.

Analysis
In Harry Potter and the Philosopher's Stone, Rubeus Hagrid says "There isn't a one of them (wizards) that turned out bad that wasn't from Slytherin." His statement is somewhat hyperbolic, as there is a counter-example in Peter Pettigrew. Pettigrew was a friend of James Potter, who was in Gryffindor House, and it can be assumed at that stage, and is made explicit later on, that he, Sirius Black and Remus Lupin were also Gryffindors. Hagrid, it is true, did not know that at the time: it was unknown that Peter had turned to the dark side, it appeared he had died at the hands, or wand, of Sirius Black while he was attempting to somehow capture Black, and that Black had been responsible for the betrayal; but then Black himself was a Gryffindor, so Hagrid even with his knowledge should have laid the blame on at least one Gryffindor. Of course, Hagrid might have got confused because of Sirius's full-fledged Slytherin family history. We will note that the revelation of Sirius as the supposed agent of the Potters' destruction, which occurs the Three Broomsticks pub in Harry Potter and the Prisoner of Azkaban comes as a surprise to Hagrid. It is possible that Hagrid's hyperbole was inadvertent, he might simply never have thought about why Pettigrew had attacked Sirius.

Questions

 * 1) Will Pettigrew be an asset to Harry in Harry Potter and the Deathly Hallows?
 * 2) Will Pettigrew be true to his debt?
 * 3) As Gryffindor house favours courage and bravery above all other attributes, why was Pettigrew chosen for this house? At no point in the series does he demonstrate these features, but rather the opposite, so why did the Sorting Hat place him in Gryffindor?

Greater Picture
Many questions have been raised about Peter Pettigrew's actions. No full picture is ever given of his pathway from a member of the Order of the Phoenix to a Death Eater, in contrast to Snape who went the opposite journey and whose story and motives are eventually told. Although Voldemort never trusted him, Pettigrew was the first to return to him, and Voldemort has since chosen him to execute some of his more sensitive missions. So, the question should be asked, during many pivotal events in the series, what was Pettigrew doing?

Let us look at the events on the day the Harry's parents were murdered. Voldemort's intent was to kill Harry, at which he failed. But the killing curse he used is harmless to chattels, only killing people, as we can easily see in the Riddle mansion which remained totally undamaged despite the three people within it being murdered by the same curse. (While curses of power, like the Killing Curse, do apparently cause explosions when they miss, we have to assume that Voldemort generally didn't miss.) And yet, when Sirius Black recounts that day, he states that he saw "the destroyed house... and their bodies" – though possibly not Voldemort's body. And why would the house be destroyed? Would Voldemort have traveled to Godric's Hollow alone? Or would he have brought a lieutenant along? Who better than Pettigrew, who had delivered the secret to Voldemort? And what would Pettigrew have done if his master failed? After the third curse, there is a silence... a silence so long that finally, Pettigrew enters the house to see if something has gone wrong. There, he finds his master apparently dead... and, quite possibly, goes briefly mad, destroying everything that he can lay his wand on, but avoiding Harry, who somehow managed to kill Voldemort. He then departs, carrying Voldemort's body (possibly) and wand (probably), and hides them; he then confronts Sirius, and using his own wand, blows up the street and makes good his escape, leaving Sirius to take the blame.

Later information is inconclusive on this subject. The damage to the Potter house is not as extensive as we are led to believe, and we do not know whether there was a body to remove or not. As Voldemort's body remains after his Killing Curse rebounds in the last chapter of the final book, it is very likely that his body was also present in this instance. We also note that there is no explosion in either of the two cases where Voldemort's killing curse rebounds upon him in the final book, he simply quietly collapses. It is reasonably certain that Peter did recover the wand, at the least, as he had it when he was nursing the returned Voldemort. It is also entirely possible that, finding his master dead, Pettigrew blasted his way out through the house's side, that being the quickest way to get himself, and possibly Voldemort's body, out of there. In Voldemort's memories, as recounted in Harry Potter and the Deathly Hallows, Voldemort recalls escaping through the "ruined house"; but as discussed elsewhere, it is entirely possible that at that point it is only his memories or senses that have been ruined, as he is then a disembodied spirit.

Twelve years pass, during which Pettigrew hides himself within a Wizarding family, the Weasleys, so that he can remain in touch with the Wizarding World's news. This period ends in Harry Potter and the Prisoner of Azkaban. As a rat cannot wield a wand, Voldemort's wand must have been concealed somewhere during this time.

Between Harry Potter and the Prisoner of Azkaban and the start of school in Harry Potter and the Goblet of Fire, Pettigrew locates and rejoins Voldemort, retrieving Voldemort's wand first, while also fetching Bertha Jorkins to take her to Voldemort, then assists in her murder; carries Voldemort to Riddle Manor, where Frank Bryce is murdered, then to Barty Crouch Jr.'s house, and assists in recruiting him and placing Bartemius Crouch Sr. under the Imperius curse; and tends to Voldemort's immediate physical needs. During Harry Potter and the Goblet of Fire, we know that Pettigrew is in Bartemius Crouch Sr.'s house, along with Voldemort and Bartemius. Voldemort will not let Pettigrew leave, as he still needs too much care. (Of course, once Bartemius escapes, Voldemort has to return to Little Hangleton, and Pettigrew will travel with him.) But we rarely see him during Harry Potter and the Order of the Phoenix. In particular, at the Battle in the Ministry, he is conspicuous by his absence. Many escaped Death Eaters are there, as is Voldemort himself. Where is Pettigrew? Is he somewhere within the Ministry that can be reached by a rat? And while he is at the house in Spinner's End at the beginning of Harry Potter and the Half-Blood Prince, where is he through the rest of that book? He cannot follow Snape to Hogwarts. What tasks might he be doing for Voldemort?

We also have one outstanding issue: wands. While tending to Voldemort, Pettigrew apparently uses Voldemort's wand. But once Voldemort has returned fully, he needs his own wand. What wand can Pettigrew use? He cannot just buy one. If it is suspected that he is actually alive, he is certainly a "person of interest" and would be arrested on sight. His original wand was returned to his family, presumably, along with his severed finger. As he must conceal that he is alive, that wand is unavailable. What use to Voldemort is a wizard without a wand, particularly a weak one like Pettigrew? Could this have something to do with Ollivander's disappearance?

As it turns out, we never learn what, if any, mission Pettigrew has during Harry Potter and the Order of the Phoenix or Harry Potter and the Half-Blood Prince. He seems to be performing menial duties for Voldemort in Harry Potter and the Deathly Hallows. And though Ollivander is required to make him a new wand, that is more of a side issue; Voldemort actually captured Ollivander for information regarding why Harry's and Voldemort's wands behaved as they did during the graveyard duel.

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


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User:Mysidia

Admittedly, not much to see here, just yet.

My user page on Wikipedia: Wikipedia:User:Mysidia

title: User:Mysidia


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User:EatAlbertaBeef

About Me
I'm a 15 year old student living in Calgary.

My Articles Are Biased
I use Wikipedia for everything, and I feel that I should contribute to the website past spelling mistakes and general riff raff. So whatever article I write i'd appreciate it if somebody would look over it and make the proper corrections, etc.

My Political Views
I'm a moderate conservative, described as a centralist by many. Even if I don't have a direct alignment it doesn't change the fact that i'm right. I think its extremelly stupid every userpage I see on this website talks about some "valiantly fighting against the right-wing radicals" or some tripe. Well i'm stating very blatently. I will stand up for the right. I support the Conservative Party of Canada and have great resentment towards the Liberal Party and the Western Alienation we face out here. My name is a reference to the "Mad Cow Crisis".

title: User:EatAlbertaBeef


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Cookbook talk:Limoncello

The measurements here are questionable and the "prod" part of "time" makes no sense to me. Kellen T 03:46, 6 August 2005 (UTC)

Agreed - how does this render only one or two servings?! -- (talk) 06:37, 9 July 2008 (UTC)

Vodka?
Vodka? Why use vodka for limoncello? I mean, never seen this here in Italy. Well, world is beautiful 'cause is varied —The preceding unsigned comment was added by 213.140.17.109 (talk • contribs) 2007-06-28T08:06:38.

 * Good point. Limoncello can be made from a variety of alcohols, but some of the best is made with pure spirit - like the Everclear sold in the USA - and diluted back as appropriate. It can also be made with grappa, for a darker limoncello with other flavour contributors. Would you care to update the recipe with your thoughts? Webaware talk 23:32, 27 June 2007 (UTC)

Webtender
The recipe seems to have been taken wholesale from Webtender. They do not claim copyright on individual recipes, so it's not a copyvio that needs to be removed, but it still seems like it would be a good idea to adapt it better to this site. Improving the grammar would be a good start. As well as changing the recipe to better match the classic method, as the comment above indicates. Ben Brockert 10:32, 14 July 2007 (UTC)

title: Cookbook talk:Limoncello


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Diagnostic Radiology/Musculoskeletal Imaging/Dysplasia Basic/Neurofibromatosis

Neurofibromatosis type I (NF-1), along with Neurofibromatosis type II (a.k.a. MISME syndrome), Tuberous Sclerosis, Sturge-Weber, and Von Hippel-Lindau compromise the phakomatosis or neurocutaneous syndromes, all of which have neurologic and dermatologic lesions. This grouping is an artifact of an earlier time in medicine, before the distinct genetic basis of each of these diseases was understood.

Etiology
Neurofibromatosis type I (NF-1), also known as von Recklinghausen disease, is the result of a defect on Chromosome 17.

Prevalence
1 in 3000-4000 people. Half are caused by new mutations (mutation rate is 1 case per 10,000 population).

Inheritance Pattern

 * Autosomal dominant
 * 100% penetrance

Mortality/Morbidity

 * Mean age of death is 54.4 years, compared to 70.1 for the general US population.
 * Malignant transformation of a peripheral nerve sheath tumor is 3-15%.

Diagnostic Criteria

 * Need at least 2
 * > 6 cafe au lait spots (larger than 5 mm in children, larger than 15 mm in adults)
 * > 2 pigmented iris hamartomas (Lisch nodules)
 * Axillary, inguinal freckling (Crowe sign)
 * > neurofibroma (or 1 plexiform neurofibroma)
 * optic nerve glioma
 * first-degree relative with NF1
 * dysplasia of greater wing of sphenoid

Radiologic Findings
Radiologic Findings:

 * Musculoskeletal: Sphenoid dysplasia (empty orbit sign), pseudarthrosis, ribbon rib deformity, bowing of long bones
 * Spine: Kyphoscoliosis in thoracic spine, vertebral scalloping, enlarged neural foramina, lateral thoracic meningocele
 * Nerve sheath tumors: neurofibromas (dumbbell lesion), plexiform neurofibroma, malignant peripheral nerve sheath tumor
 * GI tract: neurofibromas in stomach and bowel which may lead to obstruction/intussusception, small bowel leiomyoma, small bowel malabsorption
 * Eye: Lisch nodules, optic glioma
 * Skin: cafe-au-lait spots, cutaneous and subcutaneous neurofibromas, freckling

Fig. 1a and b Two images from two different patients demonstrates extensive neurofibromas involving bilateral sciatic nerves (image a), and exiting out the sciatic notch into the peripheral nerves of the right gluteal and thigh muscles (image b).

Fig. 2 Two images from the same patient shows a large paravertebral soft tissue mass engulfing C6-8 nerve roots. Axial image demonstrates tumor extending out the left neural foramen.

Fig. 3 Two images from the same patient demonstrates large bilateral neurofibromas involving bilateral lumbar nerve roots, and enlarging the neural foramen. Axial image demonstrates the so-called dumbbell lesion.


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Muggles' Guide to Harry Potter/Places/Hogsmeade

General Overview
Hogsmeade is the wizarding village near Hogwarts School of Witchcraft and Wizardry. The Hogwarts Express stops at Hogsmeade village. It is the only purely Wizard occupied town in Britain.

Extended Description
While we first hear of Hogsmeade in the first book, Harry Potter and the Philosopher's Stone, at that point it is simply a passing reference to Hogsmeade Station, the terminus of the Hogwarts Express. That there is an actual town there, and the significance of that town, is not explored until the third book, Harry Potter and the Prisoner of Azkaban

Located near Hogwarts School of Witchcraft and Wizardry, Hogsmeade is "the only non-Muggle settlement in Britain", and the northern terminus of the Hogwarts Express train line from London. It was founded by Hengist of Woodcroft, who was fleeing because of the Muggles' oppression, at about the same time as Hogwarts was founded. Hogwarts students that attend the third year or above may, on selected weekends (so-called Hogsmeade weekends) and with parental permission, visit the town and patronize its many shops.

A memorable feature of Hogsmeade is the Shrieking Shack, infamous as the "most severely haunted building in Britain".

Hogsmeade merchants

 * Dervish and Banges, selling and repairing magical items or equipment

 * Gladrags Wizardwear, clothes shops
 * with shops in London

 * The Hog's Head, a small inn off the main street

 * Honeyduke's Sweetshop, selling a wide variety of candies, fudges and blocks of chocolates
 * owned by Ambrosius Flume and his wife, who both live over the shop

 * a post office
 * has three hundred owls, including some very small Scops owls for "Local Deliveries Only"

 * Madam Puddifoot's tea shop
 * owned by Madam Puddifoot
 * couples of Hogwarts students are drawn to this shop for an idyllic tête-à-tête

 * Scrivenshaft's Quill Shop, selling quills and stationery

 * The Shrieking Shack, the "most severely haunted building in Britain"

 * The Three Broomsticks, a pub that also caters to the under-age students
 * owned by Madam Rosmerta
 * A favorite drink among the students is butterbeer. Other drinks include gillywater, mulled mead, red currant rum, cherry syrup and soda with ice and an umbrella, and firewhisky for the mature drinkers.

 * Zonko's Joke Shop, selling items that can be used for jokes and tricks
 * favorite shop of Fred and George Weasley
 * the items sold include Dungbombs, Hiccup Sweets, Frog Spawn Soap, and Nose-Biting Teacups, among other things
 * in book 6, this shop closes for lack of business when Hogwarts cancels further Hogsmeade weekends

Analysis
As mentioned, Hogwarts students in their third year and later, with parental permission, may visit Hogsmeade on selected weekends. These Hogsmeade weekends, occurring every month or two, are eagerly anticipated by all eligible students, and the school is practically vacant on those days. Harry is particularly dismayed when he is unable to secure permission from Uncle Vernon, Cornelius Fudge, or Professor McGonagall, and must remain behind while Ron and Hermione head off to sample the joys of the village. Harry does eventually get to the village with assistance from Fred and George, and later is granted permission by his godfather, Sirius Black.

In the article on Prefects, it is noted that Hogwarts is run on a fairly typical boarding school model. In a boarding school situation, the Town visits typified by the Hogsmeade weekend are used as a way of relieving pressure on the students. The school, acting in loco parentis for the students, is responsible for them, and therefore must be able to control them at all times. However, keeping them entirely on the school grounds makes school feel like a prison to the students. Thus, allowing the occasional visit off the school property is a release of tension for the students, and likely for the teachers as well. In the series, it allows the introduction of characters from outside the walls of the school, such as Madam Rosmerta, and to a certain extent also allows stories from outside the school to reach the students. Additionally, it provides a place where students can meet outside the confines of the school buildings, and thus supposedly away from observation by teachers.

In Harry Potter and the Half-Blood Prince, the town seems to be falling on hard times; in particular, following Katie Bell being injured by a cursed necklace she apparently got in Hogsmeade, all Hogsmeade weekends are canceled. Apparently, Zonko's joke shop closes for lack of business, causing Fred and George to re-think their idea of opening a branch there. The lack of student visits is bound to cause hardship for many other shops in the village as well, including the Three Broomsticks, Honeyduke's, and Madam Puddifoot's, all of which are usually filled to capacity on Hogsmeade weekends.

In Harry Potter and the Deathly Hallows, the town has been placed under a sunset-to-sunrise curfew by Death Eaters. Anyone on the streets after sunset will trigger an alarm, the Caterwaul, which will summon Death Eaters from where they stand watch, apparently The Three Broomsticks. The Hog's Head, it transpires, is suffered to remain operating as a place where business can be transacted without being watched; while the Death Eaters are not happy about not being in control there, they still have business that they prefer to carry out unmonitored by their fellow wizards, Dark or otherwise. The owner of The Hog's Head is able to face down a squad of Death Eaters by offering to close up shop and leave.

Greater Picture
One less-obvious factor in the development of the story is that Hogsmeade allows students to meet with characters who are, if not entirely in the underworld of the Wizarding world, at least somewhat more shady than you would expect to find at the school. In particular, we spend some time within the confines of the seedy Hog's Head Inn pub, and meet with Mundungus Fletcher. Fletcher, and the areas of the Wizarding world in which he dwells, are necessary to make the story universe complete, and given that they have been introduced, will be vital also for Harry to complete his quest.

It is this attention to detail that to a large extent makes the Wizarding world seem so complete. It would be a simpler story if there were only those wizards aligned with Voldemort, and those arrayed against him, but the world would seem to be missing something. Additionally, by painting the world as all either black or white, the author would be diluting her own message. While we see him in many other places, it is in Hogsmeade that we see Fletcher, the opportunist, and how he maneuvers through the world, whether it is the Ministry or Voldemort who is in charge of it; and it is here also that we see the Hog's Head, that survives, not by breaking rules itself, but by turning a blind eye on those who choose to.


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Diagnostic Radiology/Musculoskeletal Imaging/Dysplasia Basic/Tarsal Coalition

Tarsal coalition is an abnormal union between tarsal bones. The union can be osseous, fibrous, or cartilaginous. This causes abnormal articulation within the mid and hind foot which produces accelerated degenerative osteoarthritis of the hindfoot and midfoot. The frequency of tarsal coalition is approximately 1% and occurs in males more than females. It is bilateral in 25% of cases. These numbers are generally accepted and based on symptomatic groups. Recent post-mortem exams have been performed to attempt to discover the "true" incidence of tarsal coalition. Solomon, et al, report an incidence of non-osseous coalitions as high as 12.7%!

Etiology
Tarsal coalition is an abnormal union between tarsal bones. The union can be osseous, fibrous, or cartilaginous. This causes abnormal articulation within the mid and hind foot which produces accelerated degenerative osteoarthritis of the hindfoot and midfoot. The frequency of tarsal coalition is approximately 1% and occurs in males more than females. It is bilateral in 25% of cases. These numbers are generally accepted and based on symptomatic groups. Recent post-mortem exams have been performed to attempt to discover the "true" incidence of tarsal coalition. Solomon, et al., report an incidence of non-osseous coalitions as high as 12.7%!

Etiologies can be congenital or acquired. Congenital tarsal coalition is the most common and results from a failure of normal tarsal segmentation. Congenital tarsal coalition classically presents in the 2nd or 3rd decade of life as a painful flatfoot deformity also known as "peroneal spastic flatfoot." Acquired tarsal coalition can follow infection, trauma, surgery or inflammatory arthritis.

Which bones are involved?

The calcaneus, talus, navicular and cuboid are commonly affected. A frequent written boards question asks: Which is the most common tarsal coalition? The answer depends on who you ask. Resnick lists them in this order with the most frequent first:

 * 1) Calcaneo-navicular
 * 2) Talo-calcaneal, middle facet
 * 3) Talo-navicular
 * 4) Calcaneo-cuboid

MSK Requisites lists them this way:

 * 1) Talo-calcaneal, middle facet
 * 2) Calcaneo-navicular
 * 3) Talo-navicular
 * 4) Calcaneo-cuboid

In general, calcaneo-navicular and talo-calcaneal are far more common than talo-navicular or calcaneo-cuboid.

Radiologic Findings
Plain films classically demonstrate a number of bony abnormalities. These include:

1. The ant-eater sign: elongation of the anterior calcaneal process on the lateral view of the foot. This represents the bony bridging of calcaneo-navicular coalition.

2. Talar beaking: dorsal beaking of the talar head may indicate underlying tarsal coalition, most commonly talo-calcaneal. However, it can also indicate Rheumatoid arthritis, DISH and acromegaly.

3. Ball-and-socket mortise joint: spherical remodeling of the talar dome. This occurs when the mortise joint gradually takes responsibility for abduction and adduction that is lost when the subtalar joint is fused in talo-calcaneal coalition.

4. Subchondral sclerosis, cortical irregularity and joint space narrowing: These are all indirect findings of fibrous or cartilaginous coalition.

Imaging Strategies
What are the imaging strategies?

Calcaneo-navicular: A special 45 degree medial oblique projection will generally provide a clear view of the ant-eater sign in a complete osseous coalition. It will also show joint space narrowing, irregularity and sclerosis in a cartilaginous or fibrous coalition.

Complete osseous

Cartilaginous or fibrous coalition

Talo-calcaneal: The Harris-Beath view, a penetrated posterior oblique image (aka, the penetrated axial view), with the patient standing on the cassette and the beam angled between 35 and 45 degrees to the cassette, is designed to reveal coalition between the sustentaculum and the talus at the middle facet.

CT is usually definitive. It will clearly show any osseous coalition. It will also demonstrate the joint space narrowing, sclerosis and irregularity of cartilaginous or fibrous coalition.

Bone scan is sensitive but less specific. It will show uptake in multiple elements of the hindfoot and midfoot because of altered mechanics and weight-bearing.

Therapy
What difference does it make? What can we do about it?

Most patients with tarsal coalition have a fixed hindfoot valgus of varying severity and some loss of the normal longitudinal arch. Peroneal muscle tightness develops and the patient complains of vague dorsolateral foot pain centered on the sinus tarsi, difficulty walking on uneven surfaces, foot fatigue and occasionally a painful limp.

There are three common therapeutic options for calcaneo-navicular coalition.
 * 1) Interval casting: The least invasive approach consists of 4 - 6 week periods of casting to relieve symptoms without surgical intervention.  If this doesn't work then an operation may be necessary.
 * 2) Resection of the calcaneo-navicular bar: Resection is followed by fat or muscle interposition to prevent re-unification.  Various degrees of success have been reported.  Approximately 70% report complete or near complete resolution of symptoms and do not need further intervention (Campbell's Operative Orthopedics).  But there is one more option for the other 30%.
 * 3) Arthrodesis: Standard triple arthrodesis, including subtalar, talo-navicular and calcaneo-cuboid joints, is performed as the definitive therapy.

And the therapies for talo-calcaneal coalition:
 * 1) Interval casting can also relieve symptoms for this type of coalition.
 * 2) Triple arthrodesis.


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Muggles' Guide to Harry Potter/Places/Grimmauld Place

General Overview
The longtime London residence of the Black family of Pureblood wizards, Number Twelve is the house where Sirius Black grew up, and which he inherited upon his mother's death, along with its mad and treacherous House-Elf, Kreacher. Decorated and furnished in gloomy, largely Slytherin themes, and filled with Dark Arts tools, tomes and objects, Number Twelve has seen use as the headquarters of the Order of the Phoenix, the covert pro-Light organization headed by Albus Dumbledore. In its pro-Light use, it is protected from discovery by being placed under the Fidelius charm, with Dumbledore as its Secret-Keeper.

Location
The house is located in London, between two Muggle houses. Being protected by a Fidelius charm, it is unplottable and cannot be accessed by intruders, either Muggles or wizards who don't know its secret. To access the house, Harry Potter stood between numbers eleven and thirteen, thinking that "the headquarters of the Order of the Phoenix may be found at number twelve, Grimmauld Place, London." The only reason this worked for him is that the note he had seen this on had been written by Albus Dumbledore, the secret-keeper for the organization; the effect of the Fidelius charm is that only those who are informed directly by the Secret-keeper can know the secret.

Exterior
Much of the time, the house is hidden to all onlookers. When it is visible, it is one in a row of similar houses, with a front door at the top of a flight of steps, badly in need of paint, and with door hardware in the shape of snakes.

Interior
Once through the front door we see "old-fashioned gas lamps all along the walls, peeling wallpaper and threadbare carpet of a long, gloomy hallway, where a cobwebby chandelier glimmered overhead and age-blackened portraits hung crooked on the walls, with a large umbrella stand that looked as though it had been made from a severed troll's leg, with a row of shrunken heads mounted on plaques on the wall." The heads belonged to house-elves. "All of them had the same rather snoutlike nose."

US readers may be confused by the description of the "parlour, on the first floor", and Hermione looking down, rather than out, at the front stoop and seeing Mundungus with his collection of cauldrons that had "fallen off the back of a broom". The house is actually laid out in the extremely vertical, UK style, with the floor naming conventional for UK buildings. Starting at the basement, then, the rooms we know about are:
 * The kitchen in the basement, as is conventional for a UK house with space for servants. Often, there is access to the kitchen from an "areaway", a sunken, paved yard, but not always, and apparently in the case of the Black house, not at all. What natural light there is in the kitchen seems to come from ground-level windows set high in the walls.
 * The main floor, in this case, containing the entrance hall.
 * The first floor, above the main floor, which contains at least the parlour, and possibly other rooms.
 * The second floor, containing at least two bedrooms, one of which is occupied by Harry and Ron.
 * An apparent third floor, containing a bedroom used by the Twins, and possibly one used by Hermione and Ginny.
 * The fourth floor, containing bedrooms originally occupied by Sirius and Regulus.
 * Attic space, in which Buckbeak lived.

It is possible that there was additionally a formal dining room on the first floor, beside the parlour, and although we never see it, there may be an additional sitting room on the main floor. It is likely that there would be servants' bedrooms in the basement, and it is possible that Sirius' and Regulus' bedrooms were designed originally as servants quarters as well. There must be at least one more bedroom, where Arthur and Molly Weasley sleep when they are in London on Order business; most likely this is on the same floor as the room shared by Harry and Ron. We are also told, in Harry Potter and the Order of the Phoenix, that Kreacher, who had been absent, was found in the attic, and had apparently been staying in Sirius' mother's old room. This suggests that Sirius' mother, Walburga, may have slept in the attic space, but the book remains mute on whether her husband also slept up there, or whether Walburga changed rooms after he passed away.

When we are introduced to the house in Harry Potter and the Order of the Phoenix, it is dirty and infested with magical vermin. The house elf Kreacher has been living there alone for at least a decade, taking orders only from the incoherent and insane portrait of Sirius' mother in the entrance hall, and has neglected his cleaning chores. The Weasleys, Harry and Hermione spend many days cleaning and removing vermin and rubbish. Eventually the house is made more habitable but remains a gloomy place. When we meet Sirius' mother in her portrait, she screams and insults the newcomers constantly, only stopping when the owner of the house, in this case Sirius, returns her to her slumber by drawing the curtains over her portrait, an act that apparently takes considerable force.

Ownership
As mentioned, the house had been inherited by Sirius, as the last Black male descendant, on the death of his mother. As Sirius had by this time betrayed the family's Blood status by consorting with Muggles, the death of Sirius' only younger brother, Regulus, apparently at the hands of Voldemort, must have been a severe blow to Walburga, and may have hastened her death. Sirius does not seem to be upset at having missed her death, which occurred while he was imprisoned.

At Sirius' death, at the end of Harry Potter and the Order of the Phoenix, the house passes to Harry, according to the terms of Sirius' will. There is some question, however, of whether the house is magically entailed, such that ownership passes to the eldest surviving Black male, or failing that, to the eldest surviving Black female, who would be Bellatrix Lestrange. At the beginning of Harry Potter and the Half-Blood Prince, Professor Dumbledore tests this by summoning Kreacher and having Harry give him an order. If the house, with Kreacher, have passed to Harry, Kreacher will be bound to follow Harry's orders despite his hatred of Harry. As it turns out, Kreacher does follow Harry's orders, so the house has passed to Harry, and presumably may continue to be used as the headquarters of the Order.

With the death of Dumbledore, the house is once again abandoned. As Severus Snape had proven to be a traitor to the Order, and was privy to its secrets, the use of the house as Headquarters was not safe. Harry, Ron, and Hermione did end up living in the house through the month of August in the seventh year of our story, but were in turn forced to abandon it when its location was revealed to another Death Eater.

With the house becoming habitable under the ministrations of the changed Kreacher, it is entirely possible that Harry ended up living there once he was finished with school.

It is perhaps worth mention that Sirius, while he was a student at Hogwarts, had liberally decorated his bedroom with items relating to Gryffindor house, and had fastened them to the wall with Permanent Sticking Charms, possibly as a way of rebelling against his parents. Regulus, on the other hand, had similarly decorated his room with Slytherin house imagery.

Analysis
It is a safe assumption that the house was built by Muggles and purchased or otherwise taken over by the Black family, because of how similar the house appears to be to its neighbours on either side. While we are not directly told of this similarity, we can infer it by the lack of comment on any appearance of strangeness, when Harry first sees the house. When Harry first sights the Burrow, he thinks how it seems to be held together by magic; the fact that he makes no similar comment regarding Grimmauld Place indicates normality of its appearance.

We are told, in Harry Potter and the Order of the Phoenix, that Harry hears the Twins laughing from their bedroom above his room. From this we infer their bedroom being directly over Harry and Ron's room. The two bedrooms at the top of the house, we are told in Harry Potter and the Deathly Hallows, belong to Sirius and Regulus. It is safe to assume in Harry Potter and the Order of the Phoenix that Sirius is in his old room, and that Regulus' room remains empty, because when the Trio want to get in they discover that it is locked and Hermione must unlock it, and if anyone from the Order was using it they likely would have left it unlocked. It is based on this that we have inferred the layout of the house, as noted above.

The street address, Grimmauld Place, is clearly based on "grim old place", an apt description of the house as it appears when we first see it. Like so many place and character names in this series, the name is clearly chosen based on the character of the house; while this might seem a bit heavy handed on the part of the author, we must recall that the books were written, at least initially, for children. We note that the author does not belabour the name's origin by having her characters comment on it.

Greater Picture
It may be worth noting, as part of the internal consistency of the story, that the house-elf Kreacher is treated as a chattel, part of the fittings of the house, by all characters except Hermione. We can't be certain what Kreacher thinks, but his reactions throughout seem to indicate that he feels that he is a part of the house, no matter how much he disapproves of those within it. He treats Hermione the same as everyone else, leading us to believe he simply does not understand Hermione's overtures. This topic is discussed in somewhat greater depth in the articles on house elves and Kreacher.


text:
Muggles' Guide to Harry Potter/Cover

Under development at Wikibooks, a site providing a collection of open-content textbooks that anyone can edit.

title: Muggles' Guide to Harry Potter/Cover


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Muggles' Guide to Harry Potter/Introduction/Beginner

Included Content
The Beginner reader is not expected to have fully explored the Harry Potter storyline. A Beginner reader is assumed to have read perhaps one or two of the books (maybe even none at all). Pages marked for Beginners include some short synopses of the book plots, brief introductions to some of the key characters in the series, and overviews of several important locations where many important events happen in the series.

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Much of the Harry Potter series has yet to be explored. A great majority of the character list as well as all of the chapter summaries are beyond the Beginner level.

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Once the reader has completed more of the Harry Potter series, the next step is the Intermediate level. The Intermediate level details much more of the storyline and is suited for a reader who has nearly completed reading the series.


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Muggles' Guide to Harry Potter/Introduction/Intermediate

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Not much is kept from the Intermediate reader. Most major content is covered - filling in the gaps is all that is left. An Intermediate reader should feel knowledgeable about the topics covered in the books, but perhaps not quite grasping the full importance of scenes and character actions, referred to as the Greater Picture which is excluded until the reading of the series can be finished.

What's Next
Once the reader has completed the entire Harry Potter series, the next step is the Advanced level. The Advanced level details the entire storyline. Advanced readers have access to the Greater Picture section seen on many pages - it connects all the pieces together.


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Muggles' Guide to Harry Potter/Introduction/Advanced

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The Advanced reader should be knowledgeable in all areas of the Harry Potter series. No topic has been left unaddressed and all commentary/analysis is understood. Every page and section is accessible, including Greater Picture, to make sure that all aspects of the plot have been covered and understood in order to realize the full magnitude of scenes and relationships in the storyline.

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Re:Suggestion
See the reply on my talk page. Kellen T 05:15, 6 August 2005 (UTC)

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Communist Manifesto
Hi, I noticed you added The Communist Manifesto. Please see What_Wikibooks_is_not for reasons why this page might be better off on WikiSource, and should probably be transwikied. The exception to this is if you intend to create an annotated version of the manifesto. Kellen T 05:21, 6 August 2005 (UTC)

 * Right after I made it I looked at the creating a new wikibook page and saw that "Original Sources should go to Wikisource." I since have changed the link from the Humanities bookshelf to a wikisource link. I'll delete The Communist Manifesto. --EatAlbertaBeef 15:59, 6 August 2005 (UTC)

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This user exist solely to prevent the impersonation of User:Geocachernemesis. GeoT 05:21, 6 August 2005 (UTC)

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High School Mathematics Extensions/Further Modular Arithmetic

Introduction
Mathematics is the queen of the sciences and number theory is the queen of mathematics. -- Carl Friedrich Gauss 1777 - 1855

In the Primes and Modular Arithmetic section, we discussed the elementary properties of a prime and its connection to modular arithmetic. For the most part our attention has been restricted to arithmetic mod p, where p is prime.

In this chapter, we start by discussing some more elementary results in arithmetic modulo a prime p, and then moving on to discuss those results modulo m where m is composite. In particular, we will take a closer look at the Chinese Remainder Theorem (CRT), and how it allows us to break arithmetic modulo m into components. From that point of view, the CRT is an extremely powerful tool that can help us unlock the many secrets of modulo arithmetic (with relative ease).

Lastly, we will introduce the idea of an abelian group through multiplication in modular arithmetic and discuss the discrete log problem which underpins one of the most important cryptographic systems known today.

Assumed knowledge In this chapter we assume the reader can find inverses and be able to solve a system of congruences (Chinese Remainder Theorem) (see: Primes and Modular Arithmetic).

Wilson's Theorem
Wilson's theorem is a simple result that leads to a number of interesting observations in elementary number theory. It states that, if p is prime then
 * $$1\cdot 2\cdot 3 \cdots (p-1) \equiv p - 1 \pmod{p}$$

We know the inverse of p - 1 is p - 1, so each other number can be paired up by its inverse and eliminated. For example, let p = 7, we consider
 * 1 × 2 × .. × 6 ≡ (2 × 4) × (3 × 5) × 1 × 6 = 6

What we have done is that we paired up numbers that are inverses of each other, then we are left with numbers whose inverse is itself. In this case, they are 1 and 6.

But there is a technical difficulty. For a general prime number, p, how do we know that 1 and p - 1 are the only numbers in mod p which when squared give 1? For m not a prime, there are more than 2 solutions to x2 ≡ 1 (mod m), for example, let m = 15, then x = 1, 14, 4, 11 are solutions to x2 ≡ 1 (mod m).

However, we can show that there can only be (at most) two solutions to x2 ≡ 1 (mod p) when p is prime. We do that by a simple proof by contradiction argument. You may want to skip the following proof and come up with your own justification of why Wilson's theorem is true.

Let p be a prime, and x2 ≡ 1 (mod p). We aim to prove that there can only be 2 solutions, namely x = 1, -1

it obvious from the above that x = 1, -1 (≡ p - 1) are solutions. Suppose there is another solution, x = d, and d not equal to 1 or -1. Since p is prime, we know d - 1 must have an inverse. We substitute x with d and multiply both sides by the inverse, i.e.
 * align = right | $$x^2 - 1 \!$$       || $$ \equiv 0 \!$$
 * $$(x - 1)(x + 1) \!$$ || $$ \equiv 0 \!$$
 * }
 * }

but we our initial assumption was that d ≠ -1. This is a contradiction. Therefore there can only be 2 solutions to x2 ≡ 1 (mod p).
 * $$(d - 1)(d + 1) \!$$ || $$ = 0 \!$$
 * align = right | $$ d + 1        \!$$ || $$ = 0 \!$$
 * align = right | $$ d            \!$$ || $$ = -1 \!$$
 * }
 * align = right | $$ d            \!$$ || $$ = -1 \!$$
 * }

Fermat's little Theorem
There is a remarkable little theorem named after Fermat, the prince of amateur mathematicians. It states that if p is prime and given a ≠ 0 then
 * $$a^{p-1} \equiv 1 \pmod{p} \!$$

This theorem hinges on the fact that p is prime. Recall that if p is prime then a ≠ 0 has an inverse. So for any b and c we must have
 * $$ ab \equiv ac \pmod{p} \!$$ if and only if $$ b \equiv c\pmod{p} \!$$

A simple consequence of the above is that the following numbers must all be different mod p
 * a, 2a, 3a, 4a, ..., (p-1)a

and there are p - 1 of these numbers! Therefore the above list is just the numbers 1, 2, ... p - 1 in a different order. Let's see an example, take p = 5, and a = 2:
 * 1, 2, 3, 4

multiply each of the above by 2 in mod 5, we get
 * 2, 4, 1, 3

They are just the original numbers in a different order.

So for any p and using Wilson's Theorem (recall: 1 × 2 × ... × (p-1) ≡ -1), we get

on the other hand we also get
 * $$a\cdot 2a\cdots (p-1)a \!$$ || $$\equiv \!$$ ||$$1\cdot 2 \cdots (p-1)$$
 * || $$\equiv \!$$ ||$$-1 \!$$
 * }
 * }

Equating the two results, we get
 * $$a\cdot 2a\cdots (p-1)a \!$$ || $$\equiv \!$$ || $$a^{p-1}(1\cdot 2 \cdots (p-1))$$
 * || $$\equiv \!$$ || $$-a^{p-1} \!$$
 * }
 * }

which is essentially Fermat's little theorem.
 * align = right |$$-a^{p-1} \!$$ || $$ \equiv \!$$ || $$ -1 \!$$
 * }

*Chinese Remainder Theorem revisited*
This section is rather theoretical, and is aimed at justifying the arithmetic we will cover in the next section. Therefore it is not necessary to fully understand the material here, and the reader may safely choose to skip the material below.

Recall the Chinese Remainder Theorem (CRT) we covered in the Modular Arithmetic section. In states that the following congruences
 * $$x \equiv b \pmod{n_1} \!$$
 * $$x \equiv c \pmod{n_2} \!$$

have a solution if and only if gcd(n1,n2) divides (b - c).

This deceptively simple theorem holds the key to arithmetic modulo m (not prime)! We shall consider the case where m has only two prime factors, and then the general case shall follow.

Suppose m = piqj, where p and q are distinct primes, then every natural number below m (0, 1, 2, ..., m - 1) corresponds uniquely to a system of congruence mod pi and mod qj. This is due to the fact that gcd(pi,qj) = 1, so it divides all numbers.

Consider a number n, it corresponds to
 * $$n \equiv x_n \pmod{p^i} \!$$
 * $$n \equiv y_n \pmod{q^j} \!$$

for some xn and yn. If r ≠ n then r corresponds to
 * $$r \equiv x_r \pmod{p^i} \!$$
 * $$r \equiv y_r \pmod{q^j} \!$$

Now since r and n are different, we must have either xr ≠ xn and/or yr ≠ yn

For example take $$m = 12= 2^2\times 3$$, then we can construct the following table showing the $$x_n$$, $$y_n$$ for each n (0, 1, 2 ... 11)

Note that as predicted each number corresponds uniquely to two different systems of congruences mod 22 and mod 3.
 * n || n (mod 22) || n (mod 3)
 * 0 ||align = center | 0 ||align = center | 0
 * 1 ||align = center | 1 ||align = center |  1
 * 2 ||align = center | 2 ||align = center |  2
 * 3 ||align = center | 3 ||align = center |  0
 * 4 ||align = center | 0 ||align = center |  1
 * 5 ||align = center | 1 ||align = center |  2
 * 6 ||align = center | 2 ||align = center |  0
 * 7 ||align = center | 3 ||align = center |  1
 * 8 ||align = center | 0 ||align = center |  2
 * 9 ||align = center | 1 ||align = center |  0
 * 10 ||align = center | 2 ||align = center |  1
 * 11 ||align = center | 3 ||align = center |  2
 * }
 * 6 ||align = center | 2 ||align = center |  0
 * 7 ||align = center | 3 ||align = center |  1
 * 8 ||align = center | 0 ||align = center |  2
 * 9 ||align = center | 1 ||align = center |  0
 * 10 ||align = center | 2 ||align = center |  1
 * 11 ||align = center | 3 ||align = center |  2
 * }
 * 10 ||align = center | 2 ||align = center |  1
 * 11 ||align = center | 3 ||align = center |  2
 * }
 * 11 ||align = center | 3 ||align = center |  2
 * }

 Exercises

1. Consider m = 45 = 32 × 5. Complete the table below and verify that any two numbers must differ in at least one place in the second and third column

2. Suppose m = piqj, n corresponds to
 * $$n \equiv x_n \pmod{p^i} \!$$
 * $$n \equiv y_n \pmod{q^j} \!$$

and r corresponds to
 * $$r \equiv x_r \pmod{p^i} \!$$
 * $$r \equiv y_r \pmod{q^j} \!$$

Is it true that
 * $$n + r \equiv x_n + x_r \pmod{p^i} \!$$
 * $$n + r \equiv y_n + y_r\pmod{q^j} \!$$

and that
 * $$nr \equiv x_nx_r \pmod{p^i} \!$$
 * $$nr \equiv y_ny_r\pmod{q^j} \!$$

Arithmetic with CRT
Exercise 2 above gave the biggest indication yet as to how the CRT can help with arithmetic modulo m. It is not essential for the reader to fully understand the above at this stage. We will proceed to describe how CRT can help with arithmetic modulo m. In simple terms, the CRT helps to break a modulo-m calculation into smaller calculations modulo prime factors of m.

As always, let's consider a simple example first. Let $$m = 63 = 3^2\ \times \ 7$$ and we see that m has two distinct prime factors. We should demonstrate multiplication of 51 and 13 modulo 63 in two ways. Firstly, the standard way

Alternatively, we notice that
 * $$51\times 13 \!$$ || $$ = \!$$   || $$663 \!$$
 * || $$ = \!$$   || $$10\times 63 + 33\!$$
 * || $$\equiv \!$$|| $$33 \pmod{63} \!$$
 * }
 * || $$\equiv \!$$|| $$33 \pmod{63} \!$$
 * }
 * }
 * $$51 \equiv 6 \pmod{9} \!$$

and
 * $$51 \equiv 2 \pmod{7} \!$$

We can represent the two expressions above as a two-tuple (6,2). We abuse the notation a little by writing 51 = (6,2). Similarly, we write 13 = (4,6). When we do multiplication with two-tuples, we multiply component-wise, i.e. (a,b) × (c,d) = (ac,bd),

Now let's solve
 * $$51\times 13 \!$$ || $$ = \!$$   || $$(6,2)\times (4,6) \!$$
 * || $$ = \!$$   || $$(24,12) \!$$
 * || $$ = \!$$   || $$(2\times 9+ 6, 7 + 5) \!$$
 * || $$\equiv \!$$|| $$ (6,5) \!$$
 * }
 * || $$\equiv \!$$|| $$ (6,5) \!$$
 * }
 * }
 * $$x \equiv 6 \pmod{9} \!$$

and
 * $$x \equiv 5 \pmod{7} \!$$

we write x = 6 + 9a, which is the first congruence equation, and then

therefore we have a = 3 + 7b, substitute back to get
 * $$6 + 9a \!$$||$$\equiv \!$$||$$5 \pmod{7} \!$$
 * align=right|$$2a \!$$||$$\equiv \!$$||$$6 \!$$
 * align=right|$$ a \!$$||$$\equiv \!$$||$$3 \!$$
 * }
 * align=right|$$ a \!$$||$$\equiv \!$$||$$3 \!$$
 * }
 * $$x = 6 + 9(3+7b) = 33 + 63b \equiv 33 \pmod{63} \!$$

which is the same answer we got from multiplying 51 and 13 (mod 63) the standard way!

Let's summarise what we did. By representing the two numbers (51 and 13) as two two-tuples and multiplying component-wise, we ended up with another two-tuple. And this two-tuple corresponds to the product of the two numbers (mod m) via the Chinese Remainder Theorem.

We will do two more examples. Let $$m = 88 = 2^3\times 11$$, and lets multiply 66 and 40 in two ways. Firstly, the standard way and now the second way, 40 = (0,7) and 66 = (4,0) and

For the second example, we notice that there is no need to stop at just two distinct prime factors. We let $$m = 975 = 3\times 5^2\times 13$$, and multiply 900 and 647 (mod 975), For the other way, we note that 900 ≡ 0 (mod 3) ≡ 0 (mod 25) ≡ 3 (mod 13), and for 647 ≡ 2 (mod 3) ≡ 22 (mod 25) ≡ 10 (mod 13), now if we solve the following congruences

then we will get x ≡ 225!
 * $$x \equiv 0 \pmod{3} \!$$
 * $$x \equiv 0 \pmod{25} \!$$
 * $$x \equiv 4 \pmod{13} \!$$
 * }
 * $$x \equiv 4 \pmod{13} \!$$
 * }

Why? If anything, breaking modular arithmetic in m into smaller components seems to be quite a bit of work. Take the example of multiplications, firstly, we need to express each number as a n-tuple (n is the number of distinct prime factors of m), multiply component-wise and then solve the resultant n congruences. Surely, it must be more complicated than just multiplying the two numbers and then reduce the result modulo m. So why bother studying it at all?

By breaking a number m into prime factors, we have gained insight into how the arithmetic really works. More importantly, many problems in modular m can be difficult to solve, but when broken into components it suddenly becomes quite easy, e.g. Wilson's Theorem for a general m (discussed below).

Exercises
1. Show that addition can also be done component-wise.

2. Multiply component-wise 32 and 84 (mod 134).

...

Euler totient
To discuss the more general form of Wilson's Theorem and Fermat's Little Theorem in mod m (not prime), it's nice to know a simple result from the famous mathematician Euler. More specifically, we want to discuss a function, called the Euler totient function (or Euler Phi), denoted φ.

The φ functions does a very simple thing. For any natural number m, φ(m) gives the number of n < m, such that gcd(n,m) = 1. In other words, it calculates how many numbers in mod m have an inverse. We will discuss the value of φ(m) for simple cases first and then derive the formula for a general m from the basic results.

For example, let m = 5, then φ(m) = 4. As 5 is prime, all non-zero natural numbers below 5 (1,2,3 and 4) are coprimes to it. So there are 4 numbers in mod 5 that have inverses. In fact, if m is prime then φ(m) = m - 1.

We can generalise the above to m = pr where p is prime. In this case, we try to employ a counting argument to calculate φ(m). Note that there are pr natural numbers below m (0, 1, 2 ... pr - 1), and so φ(m) = pr - (number of n < m such that gcd(n,m) ≠ 1). We did that because it is easier to count the number of n 's without an inverse mod m.

An element, n, in mod m does not have an inverse if and only if it shares a common factor with m. But all factors of m (not equal to 1) are a multiple of p. So how many multiples of p are there in mod m? We can list them, they are
 * $$0, p, 2p, \cdots, p^r - p \!$$

where the last element can be written as (pr-1 - 1)p, and so there are $$p^{r-1} \!$$ multiples of p. Therefore we can conclude
 * $$\phi (p^{r}) = p^r - p^{r-1} \!$$

We now have all the machinery necessary to derive the formula of φ(m) for any m.

By the Fundamental Theorem of Arithmetic, any natural number m can be uniquely expressed as the product of primes, that is
 * $$m = p_1^{k_1}p_2^{k_2}\cdots p_r^{k_r} \!$$

where pi for i = 1, 2 ... r are distinct primes and ki are positive integers. For example 1225275 = 3×52×17×312. From here, the reader should try to derive the following result (the CRT may help).

 Euler totient function φ  Suppose m can be uniquely expressed as below $$m = p_1^{k_1}p_2^{k_2}\cdots p_r^{k_r} \!$$ then $$\phi(m) = (p_1^{k_1} - p_1^{k_1-1})(p_2^{k_2}-p_2^{k_2-1})\cdots (p_r^{k_r}-p_r^{k_r-1}) \!$$

With the Euler totient function we can derive a more general case of Fermat's Little Theorem, that is:

$$a^{\phi(m)} \equiv 1 \pmod{m} \ $$

Wilson's Theorem
Wilson's Theorem for a general m states that the product of all the invertible element in mod m
 * equals -1 if m has only one prime factor, or m = 2pk for some prime p
 * equals 1 for all other cases

An invertible element of mod m is a natural number n < m such that gcd(n, m) = 1. A self-invertible element is an element whose inverse is itself.

In the proof of Wilson's Theorem for a prime p, the numbers 1 to p - 1 all have inverses. This is not true for a general m. In fact it is certain that (m - 1)! ≡ 0 (mod m), for this reason we instead consider the product of all invertible elements in mod m.

For the case where m = p is prime we also appealed to the fact 1 and p - 1 are the only elements when squared gives 1. In fact for m = pk, 1 and m - 1 (≡ -1)are the only self-invertible elements (see exercise). But for a general m, this is not true. Let's take for example m = 21. In arithmetic modulo 21 each of the following numbers has itself as an inverse
 * 1, 20, 8, 13

so how can we say the product of all invertible elements equal to 1?

We use the CRT described above. Let us consider the case where m = 2pk. By the CRT, each element in mod m can be represented as a two tuple (a,b) where a can take the value 0 or 1 while b can take the value 0, 1, ..., or pk - 1. Each two tuple corresponds uniquely to a pair of congruence equations and multiplication can be performed component-wise.

Using the above information, we can easily list all the self-invertible elements, because (a,b)2 ≡ 1 means (a2,b2) = (1,1), so a is an invertible element in mod 2 and b is an invertible element in mod pk, so a ≡ 1 or -1, b ≡ 1 or -1. But in mod 2 1 ≡ -1, so a = 1. Therefore, there are two elements that are self invertible in mod m = 2pk, they are (1,1) = 1, and (1, -1) = m - 1. So in this case, the result is the same as when m has only a single prime factor.

For the case where m has more than one prime factors and m≠ 2pk. Let say m has n prime factors then m can be represented as a n-tuple. Let say m has 3 distinct prime factors, then all the self-invertible elements of m are their product is (1,1,1) which corresponds to 1 in mod m.
 * (1,1,1)
 * (1,1,-1)
 * (1,-1,1)
 * (1,-1,-1)
 * 1) (-1,1,1)
 * 2) (-1,1,-1)
 * 3) (-1,-1,1)
 * 4) (-1,-1,-1)

Exercise
1. Let p be a prime. Show that in arithmetic modulo pk, 1 and pk - 1 are the only self-invertible elements.

...more to come

Fermat's Little Theorem
As mentioned in the previous section, not every element is invertible (i.e. has an inverse) mod m. A generalised version of Fermat's Last Theorem uses Euler's Totient function, it states
 * $$a^{\phi(m)} \equiv 1 \pmod m \!$$

for all a ≠ 0 satisfying gcd(a,m) = 1. This is easy to see from the generalised version of Wilson's Theorem. We use a similar technique from the prove of Fermat's Little Theorem. We have
 * $$(ab_1)(ab_2)\cdots (ab_{\phi(m)}) \equiv b_1b_2\cdots b_{\phi(m)} \pmod m$$

where the bi's are all the invertible elements mod m. By Wilson's theorem the product of all the invertible elements equals to, say, d (= 1 or -1). So we get
 * $$a^{\phi(m)}d \equiv d \pmod m \!$$

which is essentially the statement of Fermat's Little Theorem.

Although the FLT is very neat, it is imprecise in some sense. For example take m = 15 = 3 × 5, we know that if a has an inverse mod 15 then aφ(15) = a8 ≡ 1 (mod 15). But 8 is too large, all we need is 4, by that we mean, a4 ≡ 1 (mod 15) for all a with an inverse (the reader can check).

The Carmichael function λ(m) is the smallest number such that aλ(m) ≡ 1 (mod m) for invertible a. A question in the Problem Set deals with this function.

Exercises
...more to come

Two-torsion Factorisation
It it quite clear that factorising a large number can be extremely difficult. For example, given that 76372591715434667 is the product of two primes, can the reader factorise it? Without the help of a good computer algebra software, the task is close to being impossible. As of today, there is no known efficient all purpose algorithm for factorising a number into prime factors.

However, under certain special circumstances, factorising can be easy. We shall consider the two-torsion factorisation method. A two-torsion element in modular m arithmetic is a number a such that a2 ≡ 1 (mod m).

Let's consider an example in arithmetic modulo 21. Note that using the CRT we can represent any number in mod 21 as a two-tuple. We note that the two-torsion elements are 1 = (1,1), 13 = (1,-1), 8 = (-1,1) and 20 = (-1,-1). Of interest are the numbers 13 and 8, because 1 and 20 (≡ - 1) are obviously two-torsion, we call these numbers trivially two-torsion.

Now, 13 + 1 = (1,-1) + (1,1) = (2,0). Therefore 13 + 1 = 14 is an element sharing a common factor with 21, as the second component in the two-tuple representation of 14 is zero. Therefore GCD(14,21) = 7 is a factor of 21.

The above example is very silly because anyone can factorise 21. But what about 24131? Factorising it is not so easy. But, if we are given that 12271 is a non-trivial (i.e. ≠ 1 or -1) two-torsion element, then we can conclude that both gcd(12271 + 1,24131) and gcd(12271 - 1,24131) are factors of 24131. Indeed gcd(12272,24131) = 59 and gcd(12270,24131) = 409 are both factors of 24131.

More generally, let m be a composite, and t be a non-trivial two-torsion element mod m i.e. t ≠ 1, -1. Then
 * gcd(t + 1,m) divides m, and
 * gdc(t - 1,m) divides m

this can be explained using the CRT.

We shall explain the case where m = pq and p and q are primes. Given t is a non-trivial two-torsion element, then t has representaion (1,-1) or (-1,1). Suppose t = (-1,1) then t + 1 = (-1,1) + (1,1) = (0,2), therefore t + 1 must be a multiple of p therefore gcd(t,m) = p. In the other case where t - 1 = (-1,1) - (1,1) = (-2,0) and so gcd(t - 1,m) = q.

So if we are given a non-trivial two-torsion element then we have effectively found one (and possibly more) prime factors, which goes a long way in factorising the number. In most modern public key cryptography applications, to break the system we need only to factorise a number with two prime factors. In that regard two-torsion factorisation method is frightening effectively.

Of course, finding a non-trivial two-torsion element is not an easy task either. So internet banking is still safe for the moment. By the way 76372591715434667 = 224364191 × 340395637.

Exercises
1. Given that 18815 is a two-torsion element mod 26176. Factorise 26176.

...more to come'

Next Section
Next Section: Multiplicative Group

Problem Set Problem Set


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User:Yosofun~enwikibooks

A bit more info on my wikipedia userpage... but then again, I probably still have to put more content on that one, too. You can discern (some of) my interests by the stuff that I contribute to. From that, you can paint your own mental image of me.

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title: User:Yosofun~enwikibooks