A math amateur can still be published

Lecture on the Fermat's conjecture

Slides for the 30-minute lecture on the Fermat Hypothesis, which I gave in July 2008 in the science tent of the University of Bonn: pdf.

The individual slides with notes

Below are many links to the German Wikipedia pages. It is often worthwhile to take a look at the corresponding English-language page (which is always linked on the left-hand side of the German-language page).

Fermat recorded his conjecture as a marginal note around 1640 in his edition of Diophant's Arithmetika.

A key point in Fermat's note, which has contributed to his "conjecture" captivating mathematicians ever since, is certain that he claims in the last two sentences that he has a proof of this statement that does not fit within this margin.

Fermat did not work full-time in mathematics, but as an amateur, but made various important contributions to mathematics: not only to number theory, but also, for example, as a co-founder of probability theory. However, his methods did not go beyond "school mathematics" (but of course he was very adept at using them).

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The name "Pythagorean number triplet" alludes to the Pythagorean theorem: for each such number triplet we get a right-angled triangle whose side lengths are the three given numbers, which in particular has whole-number sides. The Babylonian cuneiform tablet Plimpton 322 (approx. 1700 BC) already contains a list of 15 Pythagorean triplets (with quite large numbers, for example: 3367, 11018, 11521).

Pythagoras (450 BC) had already discovered a special case of the given general formula and thus given an infinite number of triples, so to speak. Euclid (300 BC) knew the formulas given (as well as the Indian mathematician Brahmagupta (600 AD)

To the given formula one should say more precisely that under the additional condition that the numbers u, v are coprime, and exactly one of them is even, just all of them primitivePythagorean triples, i.e. those in which x and y are coprime.

This case has already been dealt with by Fermat, with the evidence outlined here. He called this method the method of infinite descent. Note the tricky approach of showing a stronger assertion than the one actually desired, which is what makes the "induction step" possible.

It took Euler about 100 years to do the case n = 3 the Fermat's conjecture proved, and about another 100 years until the falls n = 5 and n = 7 were treated.

From the Mordell conjecture (which was proven by Faltings in 1983) it follows that it is for solid n can give at most a finite number of coprime solutions to the Fermat equation.

Wiles announced at a conference in Cambridge in 1993 that he had a proof of the Shimura-Taniyama-Weil conjecture, from which the Fermat conjecture follows (see below). It then turned out, however, that his suggestion of proof contained an error. However, Wiles succeeded (with the help of Richard Taylor) in modifying the evidence strategy somewhat and finally in submitting correct evidence. This was published in the Annals of Mathematics in 1995.

Since Wiles had just passed the age limit of 40, he did not receive the Fields Medal (the "Nobel Prize for Mathematicians"). But he received a variety of other prizes (for example the Schock Prize, the Cole Prize, the Wolf Prize, the Wolfskehl Prize and the Shaw Prize) and was knighted to the Knight of the British Empire.

Wiles was not a nobody in the field of number theory when he presented his proof, on the contrary: he had been a professor at the renowned Princeton University for over 10 years.

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The equation given is called an elliptic curve. These equations are of enormous importance in algebraic geometry and number theory. One of the most important open conjectures in number theory, the conjecture of Birch and Swinnerton-Dyer, concerns the question of how one can read from such an equation how many rational solutions it has (since there are usually infinite, one must still specify what you mean by that).

In the variant one considers - in a certain sense - approximate solutions of the given equation. On the one hand, this has the effect of making it easier to find solutions. On the other hand, it is sufficient for chosen p the indeterminate only in the set of numbers between 1 and p to vary so that one can count the number of solutions.

The Shimura-Taniyama-Weil conjecture was proven by Wiles (and Taylor) for a large class of elliptic curves (large enough to apply Frey's idea). Meanwhile, the work has been fully proven by a work by Breuil, Conrad, Diamond, and Taylor.

Frey had his idea in 1985. These (hypothetical) elliptical curves had also previously been considered by Hellegouarch. Ribet proved in 1986 that Frey was right.

On this more or less arbitrarily selected page from Wiles' work one sees several characteristics: instead of explicit calculations one has to do with a lot of text: arguments are explained. An extensive mathematical symbol language is used, which is only accessible to specialists.

Even if Wiles isolated himself in an unusual way from the mathematical professional world during the time he was working on his proof, it is also clear in his work that he uses the preparatory work of a large number of other mathematicians. His essay contains direct references to 84 other articles, and thus to thousands of pages of modern mathematics, most of which arose after 1950.

The images show a French and a Czech postage stamp, which are dedicated to Fermat's problem and represent here what a great response Wiles’s solution to the problem has received in public. For example, the New York Times brought several articles about it, some on the front page.

Why was there so much public response? One important reason was that the Fermatsche Conjecture is so easy to explain that anyone can understand it, and yet poses such a difficult problem. As a result, many math amateurs have been looking for a proof of this statement over the past 350 years. This interest was fueled by the Wolfskehl Prize (see below).

This great public interest in Wiles' breakthrough was also justified from the point of view of the mathematician community, albeit less because it was now established that the Fermat equation actually has no non-trivial solutions, but because the Shimura-Taniyama-Weil- Conjecture was now proven - this has a much higher relevance in number theory - and because Wiles has developed a number of new and highly interesting methods that now also allow the study of other similar problems, and also raise fascinating new questions. Last but not least, Wiles' result fits into the so-called Langlands program, a building of conjectures in algebraic number theory and representation theory that has significantly influenced the development of these disciplines over the past 50 years.

Other questions that could be discussed in such a lecture:

Did Fermat Have Any Proof? It is clear that Fermat could not have had the evidence known today in mind when he wrote his note in the margin. The most plausible assumption is surely that the reasoning he was thinking of contained an error. Of course, he must also be credited with the fact that his claim was not published by himself, but - after his death - by his son. It is even so that Fermat himself later provided evidence for his claim for fixed exponents (esp. n = 4 dealt - that would not have been necessary if he had known general evidence.

Is there an elementary proof? A similar question is whether there is any "elementary" proof of the Fermatschne conjecture.

The Wolfskehl Prize The Royal Society of Sciences in Göttingen announced a prize in 1908, which was donated by Paul Wolfskehl for the solution of the Fermat problem:

Due to the fact that the late Dr. Paul Wolfskehl in Darmstadt will be given a price of 100,000 Mk., In words: "One hundred thousand marks", for the one who first succeeds in proving Fermat's great theorem. ... If the prize is not awarded by September 13, 2007, no further claims can be made. "

The sum advertised was a fortune. The figures vary between EUR 700,000 and EUR 2.5 million over the current value. However, due to post-WWI inflation, the prize money had shrunk sharply (to around EUR 40,000) by the time Wiles received the prize. To mark the centenary, a colloquium was held in Darmstadt on June 30, 2008, at which Wiles also gave a lecture.

Open guesswork Fermat's problem is now solved, but mathematical research continues. There are many open questions and guesses; some of the most prominent are summarized in the Clay Institute's list. Solving any of the seven problems on this list will be rewarded with US $ 1 million. (The only one of these problems that has been solved so far is the Poincaré conjecture, proven by G. Perelman.)