# Great Fermatsch sentence

Pierre de Fermat

The Great Fermatsche Theorem was formulated in the 17th century by Pierre de Fermat , but only proven in 1994 by Andrew Wiles . The conclusive climax for the proof is the collaboration between Wiles and Richard Taylor , which, in addition to the final proof by Wiles, was reflected in a simultaneous publication of a partial proof by both Wiles and Taylor as joint authors.

The proposition says: If a natural number is greater than 2, then the -th power of any positive whole number can not be broken down into the sum of two equal powers: ${\ displaystyle n}$${\ displaystyle n}$

${\ displaystyle a ^ {n} + b ^ {n} = c ^ {n}}$

with positive integers is only possible for and . ${\ displaystyle a, b, c, n}$${\ displaystyle n = 1}$${\ displaystyle n = 2}$

The Great Fermatsche Theorem is considered unusual in many ways. His testimony is easy to understand even for laypeople, despite the difficulties that arose in proving it. It lasted more than 350 years and was a story of the failed attempts in which numerous leading mathematicians such as Ernst Eduard Kummer have participated since Leonhard Euler . Numerous partly romantic, partly dramatic, but also tragic episodes of this story have made him popular far beyond the circle of mathematicians.

The proof finally provided, in whose preparatory work alongside Wiles and Taylor also Gerhard Frey , Jean-Pierre Serre , Barry Mazur and Ken Ribet , is considered the high point of mathematics of the 20th century.

## Designations

There are different names for this sentence. The most common in German is Großer Fermatscher Satz and derived from it Großer Fermat in contrast to the Little Fermat Sentence or Little Fermat. Since there is no evidence from Fermat himself, it was strictly speaking initially only a conjecture. This is why the term Fermat's conjecture is also used, but Fermat's theorem was used even before the proof . In order to include Wiles, the finder of the proof, the Fermat-Wiles theorem is also mentioned . In English, the sentence is called Fermat's Last Theorem , which in German is sometimes (imprecisely) translated as Fermat's last sentence or Fermat's last theorem .

## origin

Book cover of the 1670 version of the Arithmetica des Diophantos published by Pierre de Fermat's son Clément-Samuel, with his father's comments
This page of the Arithmetica of 1670 contains Pierre de Fermat's marginal note

Probably between 1637 and 1643, an exact year cannot be given due to the circumstances explained below, Fermat wrote while reading the arithmetic of Diophantos of Alexandria next to the 8th task of the second (Greek) “book” the following lines as a marginal note in his hand copy Work:

“Cubum autem in duos cubos, aut quadratoquadratum in duos quadratoquadratos, et generaliter nullam in infinitum ultra quadratum potestatem in duas ejusdem nominis fas est dividere: cujus rei demonstrationem mirabilem sane detexi. Hanc marginis exiguitas non caperet. "

“However, it is not possible to divide a cube into 2 cubes, or a biquadrat into 2 biquadrates and generally one power, higher than the second, into 2 powers with the same exponent: I have discovered a truly wonderful proof of this, but this one is Edge here too narrow to hold it. "

Since Fermat's hand copy of the arithmetic was found by his son in his father's estate after his death and the latter did not date his notes in the margin, an exact date cannot be determined. It is plausible to assume, however, that Fermat had at least solved the case and perhaps also the case before he was tempted to make his famous and "reckless" remark. Therefore, the year 1641 is more likely than 1637. ${\ displaystyle n = 4}$${\ displaystyle n = 3}$

That Fermat had found a proof for the special case , of which he perhaps believed he could generalize, is obvious, because this special case is an easy inference from a theorem that he has explicitly proven: Area trianguli rectanguli in numeris non potest quadratus. (The area of ​​a Pythagorean triangle cannot be a square number.), Which he wrote including the proof in the margin next to the 26th task of the 6th (Greek) “book” of arithmetic. André Weil has also proven convincingly that Fermat had all the means to prove the case with his method. ${\ displaystyle n = 4}$${\ displaystyle n = 3}$

The theories used in Wiles' proof in 1995 were not even rudimentary developed over 350 years earlier. That does not exclude with certainty that one day an even simpler proof will be found that gets by with more elementary means. But most number theorists today doubt that Fermat could have found one. The surest sign that Fermat soon realized that he had not found any evidence after all is that he had not mentioned the sentence or any proof of it to any of his correspondents. Moreover, Fermat's marginal note was only intended for himself. He could not expect publication by his son Samuel.

## distribution

After Fermat's death, his number-theoretic discoveries were long forgotten because he had not had his findings printed and his contemporaries among mathematicians were not particularly interested in number theory, with the exception of Bernard Frenicle de Bessy. Fermat's eldest son, Samuel, published a new edition of Arithmetica five years after his father's death , which also included his father's forty-eight remarks. The second of these marginal notes later became known as the Fermat conjecture . Although the notes contained a number of fundamental mathematical theorems, evidence or even simple explanations of how Fermat had arrived at these results were mostly, if not all, missing. One of the most important findings of Fermat's, the famous Area trianguli rectanguli in numeris non potest esse quadratus , is provided with a complete proof in addition to the 26th task of the 6th “book” of arithmetic. Here Fermat uses his method of infinite descent . It was left to the subsequent mathematicians, above all and first to Leonhard Euler , to gradually find the missing evidence.

## uncertainty

In this context, in the centuries that followed, the (now so-called) Great Fermat's Theorem in particular developed into a challenge for many mathematicians - there was in fact no one who could prove or disprove it. But because Fermat himself had claimed the existence of a “wonderful proof”, generations of mathematicians, including the most important of their time, tried to find it. Fermat's other remarks, too, turned out to be a source of difficult, long-term work for his mathematician colleagues. All in all, however, these efforts led - almost incidentally - to a large number of significant discoveries.

## Exception n = 1, n = 2

For and has an infinite number of solutions . For the equation is simple and any solutions can be chosen. For the solutions are the Pythagorean triples . ${\ displaystyle n = 1}$${\ displaystyle n = 2}$${\ displaystyle a ^ {n} + b ^ {n} = c ^ {n}}$${\ displaystyle a, b, c \ in \ mathbb {N}}$${\ displaystyle n = 1}$${\ displaystyle a + b = c}$${\ displaystyle a, b}$${\ displaystyle n = 2}$

## Evidence for special cases of the theorem

It is enough to prove the conjecture for prime exponents and exponent 4. It is customary to distinguish two cases in the Fermat problem for a prime exponent . In the first case, solutions are sought that are not divisible by. In the second case the product divides . ${\ displaystyle p}$${\ displaystyle a, b, c}$${\ displaystyle p}$${\ displaystyle p}$${\ displaystyle a \ cdot b \ cdot c}$

Special cases of Fermat's Great Theorem could be proven early on:

### n = 3, n = 4 and multiples of these numbers

Bernard Frénicle de Bessy published the first evidence for the case as early as 1676 . His solution came from Fermat himself, of whom in this case a sketch of evidence is known in a marginal note in his Diophant edition on a closely related problem (see Infinite Descent ). ${\ displaystyle n = 4}$

Leonhard Euler published evidence of the case in 1738 . Later, with the help of the complex numbers, he was also able to confirm the claim for the case that he published in 1770 (that he had the proof, he announced in a letter as early as 1753). However, Euler did not succeed in extending his method of proof to other cases. ${\ displaystyle n = 4}$${\ displaystyle n = 3}$

At least 20 different pieces of evidence have now been found for the case . For there are at least 14 different evidence. ${\ displaystyle n = 4}$${\ displaystyle n = 3}$

### 4 and odd prime numbers are sufficient

It soon became clear that it was sufficient to prove Fermat's theorem for all prime numbers greater than 2 and for the number 4. Because every natural number that is not prime is divisible by 4 or an odd prime number. If now either 4 or an odd prime number, a natural number and as well as a solution for the exponent , there is also a solution for the exponent , namely . However, such a solution should not exist if Fermat's theorem applies to the exponent . Thus Fermat's theorem also applies to the exponent . ${\ displaystyle n> 2}$${\ displaystyle e}$${\ displaystyle d}$${\ displaystyle n = de}$${\ displaystyle a ^ {n} + b ^ {n} = c ^ {n}}$${\ displaystyle n}$${\ displaystyle e}$${\ displaystyle \ left (a ^ {d} \ right) ^ {e} + \ left (b ^ {d} \ right) ^ {e} = \ left (c ^ {d} \ right) ^ {e} }$${\ displaystyle e}$${\ displaystyle n}$

With the evidence in the cases and Fermat's last theorem was for all that a multiple is 3 or 4, proved. ${\ displaystyle n = 3}$${\ displaystyle n = 4}$${\ displaystyle n}$

The problem is that the prime numbers also represent an infinitely large set of numbers and therefore -  per se  - an infinite set of cases to be proven: With these methods a (further) plausibility check could and can be achieved, but never a conclusive and mathematically exact proof.

### n = 5, first case and Sophie Germain primes

In 1825 Peter Gustav Lejeune-Dirichlet and Adrien-Marie Legendre were able to prove the theorem for . They relied on the preparatory work of Sophie Germain . Germain was able to prove that the first case of Fermat's conjecture applies to all Sophie-Germain prime numbers (which also have a prime number). Legendre was able to extend this to the cases in which the exponent is also prime, with . This then provided the validity of the first case of the Fermat conjecture for prime numbers . However, until the work of Wiles and Taylor, no general evidence was known for either the first case ( ) or the second case ( ). However, Roger Heath-Brown , Leonard Adleman and Étienne Fouvry were able to show in 1985 that the first case of Fermat's conjecture applies to an infinite number of prime numbers, and in the first case criteria were derived that made it possible for example for Andrew Granville in 1988 to prove that the first Part of the presumption is true. ${\ displaystyle n = 5}$${\ displaystyle p}$${\ displaystyle 2p + 1}$${\ displaystyle n = p}$${\ displaystyle k \ cdot p + 1}$${\ displaystyle k = 4,8,10,14,16}$${\ displaystyle p <100}$${\ displaystyle n \ nmid abc}$${\ displaystyle n \ mid abc}$${\ displaystyle p}$${\ displaystyle p <6 {,} 93 \ cdot 10 ^ {17}}$

### n = 14 and n = 7

Dirichlet was able to provide evidence for the case in 1832 . In 1839, Gabriel Lamé showed that the case is also valid. Like Augustin-Louis Cauchy , Lamé was still convinced in March 1847 that complete proof of Fermat's conjecture could be presented to the French Academy of Sciences within weeks (hope was destroyed a little later by a letter from Kummer). ${\ displaystyle n = 14}$${\ displaystyle n = 7}$

Later, simpler variants of the proof were also found. ${\ displaystyle n = 7}$

### Further individual cases

In 1885, GB Matthews presented evidence of the cases and . J. Fell published an article in 1943 in which he a method for setting out, also for and should be applicable. ${\ displaystyle n = 11}$${\ displaystyle n = 17}$${\ displaystyle n = 11}$${\ displaystyle n = 17}$${\ displaystyle n = 23}$

### All regular prime numbers

However, the hope of a quick (and general) proof expressed by Cauchy and Lamé in 1847 was dashed by Ernst Eduard Kummer , who discovered a reasoning error in Lamé and Cauchy's considerations: they had tacitly assumed that in the whole end of the whole numbers in the The expansions of the field of rational numbers considered by them ( fields of circular division of the order ) for the respective Fermat equation for the exponent (it arises from the adjunction of the -th roots of unity) the unambiguous prime factorization still applies. ${\ displaystyle p}$${\ displaystyle p}$${\ displaystyle p}$

Kummer developed a theory in which the unambiguous prime factorization could be saved by combining certain sets of numbers of the number field ( ideals ) and examining the arithmetic of these new "ideal numbers". He was able to prove Fermat's great theorem in 1846 for regular prime numbers ; A prime number is called regular if none of the Bernoulli numbers have their numerator divisible by . In this case, the class number - ie the number of non-equivalent ideal classes - the cyclotomic field of order not divisible. It is not known whether there are infinitely many regular prime numbers. ${\ displaystyle p}$ ${\ displaystyle B_ {0}, B_ {2}, \ dotsc, B_ {p-3}}$${\ displaystyle p}$${\ displaystyle p}$${\ displaystyle p}$

With the help of the computer and further development of Kummer's methods, Harry Vandiver succeeded in proving the theorem for all prime numbers less than 2000 as early as the 1950s. With the help of the computer, the limit could be shifted considerably upwards, but one did not get any closer to proving Fermat's conjecture in this way, it only became more plausible.

### At most finitely many coprime solutions for n ≥ 4 with fixed n

From Mordell's conjecture - proved in 1983 by Gerd Faltings  - it follows as a special case that if one of Fermat's equations has for a solution, it can only have at most a finite number of coprime solutions. ${\ displaystyle n \ geq 4}$

## Wolfskehl Prize

The search for general evidence was also materially motivated at the beginning of the 20th century by the will of the Darmstadt doctor and mathematician Paul Friedrich Wolfskehl . According to a legend told later , his fate was strangely linked to Fermat's theorem. When his love for a woman was not returned by her, he made up his mind to kill himself. He set the time of his suicide at exactly midnight. In order to bridge the time until his suicide, he reread one of Ernst Eduard Kummer 's relevant works on Fermat's conjecture and believed that he had found a mistake in it. He began to check it out carefully and forgot the time. By the time Wolfskehl finally found out that Kummer hadn't made a mistake, the planned time of his suicide was already over and he decided to give up on his plan. In gratitude that Fermat had effectively saved his life, he changed his will. - But that is an unproven legend.

When he died of natural causes ( multiple sclerosis ) in 1906 , it was announced that in his last will he had offered a price of 100,000 gold marks for whoever  would first publish complete evidence in a professional journal . In 1908 the Royal Society of Sciences in Göttingen announced the Wolfskehl Prize . The closing date for this project should be September 13, 2007. In 1997 the prize, which was still worth 75,000 DM, was paid to Andrew Wiles .

## The proof

In 1993 Andrew Wiles presented a proof of the Taniyama-Shimura conjecture in lectures at the Isaac Newton Institute in Cambridge , which would also prove Fermat's great theorem. However, the evidence he presented was incomplete in one essential point, as only became apparent in the subsequent review . Together with his student Richard Taylor , Wiles was able to close the gap in 1994 and thus also prove Fermat's great theorem.

The core of the 98-page work without appendix and bibliography consists of a two-part proof by contradiction :

• Are with a counter example to Fermat's theorem, so is the elliptic curve${\ displaystyle a, b, c, n}$${\ displaystyle a ^ {n} + b ^ {n} = c ^ {n}}$
${\ displaystyle y ^ {2} = x \ cdot (xa ^ {n}) \ cdot (x + b ^ {n})}$
not modular . This was suspected by Gerhard Frey in 1986 and proven by Ken Ribet in a 1990 contribution by Jean-Pierre Serre .

This is a contradiction to the first part of the proof, the assumed existence of a counterexample to Fermat's great theorem must be wrong.

## Conjectures that include the Fermat conjecture

There are a few open conjectures that include the Fermat conjecture as a special case, the most important being the abc conjecture . Others are the Fermat-Catalan conjecture and the Andrew Beal conjecture . Another generalization is Euler's conjecture , which has now been refuted.

## Trivia

• In the episode Hotel Royal of the television series Spaceship Enterprise: The Next Century from 1989 it is claimed that the Fermatsche theorem could not be proven even with computer help until the 24th century. Shortly after the series was discontinued in 1994, the proof was then provided. However, Star Trek generally plays in a different timeline . In 1995 there was a "correction" by the "Star Trek" authors: In the episode Facetten (season 3, episode 25) of the television series Star Trek: Deep Space Nine , the symbiote Dax is an alternative ("more original") proof of gossip And at this point explicitly referred to Andrew Wiles's solution.
• In the episode In the Shadow of the Genius of the Simpsons , Homer Simpson writes a supposed counterexample for the great Fermat sentence on a blackboard: the expression in which the difference between the two sides appears as zero in simple pocket calculators. However, this is of course not an actual solution, but only a consequence of the limitations of such a pocket calculator: Since all three numbers have the order of magnitude , but the difference between the two sides is only the comparatively small order of magnitude , the calculator can no longer resolve this. The episode Die Panik-Amok-Horror-Show also contains a supposed solution with the equation , in which the difference between the two sides is 10 orders of magnitude smaller than the numbers. Behind these mathematical contributions to the series is the author David X. Cohen , one of several members of the writing staff with a mathematical and scientific background.${\ displaystyle 3987 ^ {12} + 4365 ^ {12} = 4472 ^ {12}}$${\ displaystyle 10 ^ {43}}$${\ displaystyle 10 ^ {33}}$${\ displaystyle 1782 ^ {12} + 1841 ^ {12} = 1922 ^ {12}}$
• The author Stieg Larsson lets his protagonist Lisbeth Salander recognize the solution to Fermat's theorem in the second volume of the Millennium Trilogy , but after a head injury she cannot remember exactly.
• In the film Teuflisch , the math homework on the blackboard is to prove Fermat's Great Theorem.
• In Arno Schmidt's short novel Schwarze Spiegel , the first-person narrator solves the Fermat problem - long before Andrew Wiles: “ The black dome of the night: from the circular skylight at the zenith it came out poisonously clear and so mockingly that the snow burned eyes and soles. I sat down on the top of my two wooden steps and wrote on a large sheet: Fermat's problem: In , assuming the integer of all sizes, should never be greater than 2. I quickly proved it to myself as follows: (1)  […] The symbols swiftly pulled themselves out of the pencil, and I went on muttering cheerfully: you have to imagine that: I solve the problem of Fermat! (But the time passed by in an exemplary manner). “Unfortunately the evidence is flawed.${\ displaystyle A ^ {N} + B ^ {N} = C ^ {N}}$${\ displaystyle N}$${\ displaystyle A ^ {N} = C ^ {N} -B ^ {N}}$

## literature

### Review articles and history

• Solving Fermat. Public Broadcasting Service (PBS) television interview with Andrew Wiles.
• Paulo Ribenboim : 13 lectures on Fermat's last theorem. Springer, New York 1979 (the most important works before Wiles).
• Paulo Ribenboim: Fermat's last theorem for amateurs. Springer 2000, ISBN 0-387-98508-5 .
• Simon Singh : Fermat's last sentence. The adventurous story of a mathematical puzzle. Deutscher Taschenbuch-Verlag, Munich 2000, ISBN 3-423-33052-X .
• The Royal Society of Sciences: Announcement regarding the Wolfskehlsche Prize Foundation. News from the Royal Society of Sciences in Göttingen. Business communications. 16: 1 (1908), pp. 103-104.
• Simon Singh and Kenneth Ribet: Solving Fermat's Riddle. In: Spectrum of Science. 1/98, ISSN 0170-2971, p. 96 ff.
• CJ Mozzochi: The Fermat Diary. In: American Mathematical Society. 2000 (history of the solution from Frey).
• Kenneth A. Ribet: Galois Representations and Modular Forms. In: Bulletin of the AMS . 32 (4/1995), 375-402.
• Gerd Faltings : The Proof of Fermat's last theorem by R. Taylor and A. Wiles. (PDF; 150 kB). In: Notices of the AMS . 42 (7/1995): 743-746. An overview of the proof idea and the most important steps that is easy to understand for “beginners”.
• Peter Roquette : On the Fermat problem. (PDF; 207 kB). Lecture at the Mathematical Institute of the University of Heidelberg, January 24, 1998. Historical development up to the solution.
• Joseph Silverman , Gary Connell, Glenn Stevens (Eds.): Modular Forms and Fermat's Last Theorem. Springer-Verlag, 1997. Mathematical background material on and presentation of Wiles proof.
• Yves Hellegouarch : Invitation to the Mathematics of Fermat-Wiles. Academic Press, 2002.
• Jürg Kramer : About the Fermat conjecture. Part 1, Elements of Mathematics. Volume 50, 1995, pp. 12-25 ( PDF ); Part 2, Volume 53, 1998, pp. 45-60 ( PDF ).
• Klaus Barner: Gerhard Frey's lost letter. Mitt. Dtsch. Math Ver. 2002, No. 2, pp. 38-44.
• Takeshi Saito: Fermat's last theorem (2 volumes), Volume 1 (Basic Tools), Volume 2 (The Proof), American Mathematical Society 2013, 2014.

Commons : Large Fermatian Theorem  - collection of images, videos and audio files

## Individual evidence

1. "Last" here refers to the fact that it was the last unproven of the sentences formulated by Fermat.
2. The original is lost, but the comment can be found in an edition of Diophant's Arithmetica with translation and commentaries by Bachet and notes by Fermat, which his son edited, see Paul Tannery , Charles Henry (ed.): Œuvres de Fermat. Tome premier. Gauthier-Villars, Paris 1891, p. 291 , notes p. 434 ; after Samuel de Fermat (ed.): Diophanti Alexandrini Arithmeticorum libri sex. Bernard Bosc, Toulouse 1670, p. 61 ; For translation see Max Miller: Comments on Diophant. Academic Publishing Society, Leipzig 1932, p. 3.
3. Pierre de Fermat: Comments on Diophant. Translated from Latin and edited with annotations by Max Miller. Akademische Verlagsgesellschaft, Leipzig 1932, pp. 34–36.
4. ^ Catherine Goldstein : Un théorème de Fermat et ses lecteures. Presses Universitaires de Vincennes, Saint-Denis 1995 (French).
5. ^ André Weil: Number theory. A walk through history from Hammurabi to Legendre. Birkhäuser, Basel 1992, pp. 120–124.
6. Paulo Ribenboim: Fermat's last theorem for amateurs. Springer-Verlag, 2000, ISBN 978-0-387-98508-4 .
7. ^ André Weil: Number Theory. An approach through history from Hammurapi to Legendre. Birkhäuser, 1984, p. 76.
8. Spectrum of Science , Dossier 6/2009: The Greatest Riddles of Mathematics. ISBN 978-3-941205-34-5 , p. 8 (interview with Gerd Faltings ).
9. ^ Klaus Barner: Paul Wolfskehl and the Wolfskehl Prize. (PDF; 278 kB). In: Notices AMS. Volume 44. Number 10, November 1997 (English).
10. Peter Roquette : On the Fermat problem. (PDF; 207 kB). Lecture at the Mathematical Institute of the University of Heidelberg, January 24, 1998. Historical development up to the solution. P. 15. Retrieved August 25, 2016.
11. Simon Singh: Homer's last sentence. The Simpsons and the Math. Hanser, Munich 2013. pp. 47–54.
12. ↑ The fact that the first sum is incorrect results directly from the checksums : The bases of both summands 3987 and 4365 (like their checksums) are divisible by 3. This means that all of its powers and their sums are divisible by 3 - in contradiction to the fact that the base 4472 of this sum has a cross sum of 17 that is not divisible by 3. The falseness of the second sum can also be recognized almost without calculation, in that one deduces from the equality of the unit digits 2 of the bases 1782 and 1922 that the difference between their powers is divisible by 10, although their base is 1841.
13. ^ Arno Schmidt: Black mirrors. In: Arno Schmidt: Brand's Haide. Two stories. Rowohlt, Hamburg 1951, pp. 153-259 (first edition).