Stood unsolved for more than 350 years, until in 1994 Andrew Wiles finally. Mat’s Last Theorem, unlike the previous results which considered the Fermat. Created Date: 4/2/2006 12:02:28 AM.

1. Kenneth A Ribet

The 1670 edition of ' Arithmetica includes Fermat's commentary, particularly his 'Last Theorem' ( Observatio Domini Petri de Fermat). In, Fermat's Last Theorem (sometimes called Fermat's conjecture, especially in older texts) states that no three a, b, and c satisfy the equation a n + b n = c n for any integer value of n greater than 2. The cases n = 1 and n = 2 have been known to have infinitely many solutions since antiquity.

This theorem was first by in 1637 in the margin of a copy of where he claimed he had a proof that was too large to fit in the margin. Was released in 1994 by, and formally published in 1995, after 358 years of effort by mathematicians. The proof was described as a 'stunning advance' in the citation for his award in 2016. The proof of Fermat's Last Theorem also proved much of the and opened up entire new approaches to numerous other problems and mathematically powerful techniques. The unsolved problem stimulated the development of in the 19th century and the proof of the in the 20th century. It is among the most notable theorems in the and prior to its proof, it was in the as the 'most difficult mathematical problem', one of the reasons being that it has the largest number of unsuccessful proofs. Contents.

Overview Pythagorean origins The Pythagorean, x 2 + y 2 = z 2, has an infinite number of positive solutions for x, y, and z; these solutions are known as. Around 1637, Fermat wrote in the margin of a book that the more general equation a n + b n = c n had no solutions in positive integers, if n is an integer greater than 2.

Although he claimed to have a general of his conjecture, Fermat left no details of his proof, and no proof by him has ever been found. His claim was discovered some 30 years later, after his death. This claim, which came to be known as Fermat's Last Theorem, stood unsolved in mathematics for the following three and a half centuries. The claim eventually became one of the most notable unsolved problems of mathematics. Attempts to prove it prompted substantial development in, and over time Fermat's Last Theorem gained prominence as an. Subsequent developments and solution With the special case n = 4 proved by Fermat himself, it suffices to prove the theorem for n that are (this reduction is considered trivial to prove ). Over the next two centuries (1637–1839), the conjecture was proved for only the primes 3, 5, and 7, although innovated and proved an approach that was relevant to an entire class of primes.

In the mid-19th century, extended this and proved the theorem for all, leaving irregular primes to be analyzed individually. Building on Kummer's work and using sophisticated computer studies, other mathematicians were able to extend the proof to cover all prime exponents up to four million, but a proof for all exponents was inaccessible (meaning that mathematicians generally considered a proof impossible, exceedingly difficult, or unachievable with current knowledge). Entirely separately, around 1955, Japanese mathematicians and suspected a link might exist between and, two completely different areas of mathematics. Known at the time as the Taniyama–Shimura–Weil conjecture, and (eventually) as the, it stood on its own, with no apparent connection to Fermat's Last Theorem. It was widely seen as significant and important in its own right, but was (like Fermat's theorem) widely considered completely inaccessible to proof.

In 1984, noticed an apparent link between these two previously unrelated and unsolved problems. An outline suggesting this could be proved was given by Frey. The full proof that the two problems were closely linked, was accomplished in 1986 by building on a partial proof by who proved all but one part known as the 'epsilon conjecture' (see: and ). In plain English, these papers by Frey, Serre and Ribet showed that if the Modularity Theorem could be proven for at least the semi-stable class of elliptic curves, a proof of Fermat's Last Theorem would also follow automatically. The connection is described: any solution that could contradict Fermat's Last Theorem could also be used to contradict the Modularity Theorem. So if the modularity theorem were found to be true, then by definition no solution contradicting Fermat's Last Theorem could exist, which would therefore have to be true as well.

Although both problems were daunting problems widely considered to be 'completely inaccessible' to proof at the time, this was the first suggestion of a route by which Fermat's Last Theorem could be extended and proved for all numbers, not just some numbers. Also important for researchers choosing a research topic was the fact that unlike Fermat's Last Theorem the Modularity Theorem was a major active research area for which a proof was widely desired and not just a historical oddity, so time spent working on it could be justified professionally.

However general opinion was that this simply showed the impracticality of proving the Taniyama–Shimura conjecture. Mathematician ' quoted reaction was a common one: 'I myself was very sceptical that the beautiful link between Fermat’s Last Theorem and the Taniyama–Shimura conjecture would actually lead to anything, because I must confess I did not think that the Taniyama–Shimura conjecture was accessible to proof. Beautiful though this problem was, it seemed impossible to actually prove. I must confess I thought I probably wouldn’t see it proved in my lifetime.'

Kenneth A Ribet

On hearing that Ribet had proven Frey's link to be correct, English mathematician, who had a childhood fascination with Fermat's Last Theorem and had a background of working with elliptic curves and related fields, decided to try to prove the Taniyama–Shimura conjecture as a way to prove Fermat's Last Theorem. In 1993, after six years working secretly on the problem, Wiles enough of the conjecture to prove Fermat's Last Theorem. Wiles's paper was massive in size and scope. A flaw was discovered in one part of his original paper during and required a further year and collaboration with a past student, to resolve. As a result, the final proof in 1995 was accompanied by a second smaller joint paper showing that the fixed steps were valid.

Wiles's achievement was reported widely in the popular press, and was popularized in books and television programs. The remaining parts of the Taniyama–Shimura–Weil conjecture, now proven and known as the, were subsequently proved by other mathematicians, who built on Wiles's work between 1996 and 2001. For his proof, Wiles was honoured and received numerous awards, including the 2016. Equivalent statements of the theorem There are several alternative ways to state Fermat's Last Theorem that are mathematically equivalent to the original statement of the problem. In order to state them, we use mathematical notation: let N be the set of natural numbers 1,2,3., let Z be the set of integers 0, ±1, ±2., and let Q be the set of rational numbers a / b where a and b are in Z with b≠0. In what follows we will call a solution to x n + y n = z n where one or more of x, y, or z is zero a trivial solution. A solution where all three are non-zero will be called a non-trivial solution.

For comparison's sake we start with the original formulation. Original statement. With n, x, y, z ∈ N (meaning n, x, y, z are all positive whole numbers) and n 2 the equation x n + y n = z n has no solutions. Most popular treatments of the subject state it this way. In contrast, almost all math textbooks state it over Z:. Equivalent statement 1: x n + y n = z n, where integer n ≥ 3, has no non-trivial solutions x, y, z ∈ Z.

The equivalence is clear if n is even. If n is odd and all three of x, y, z are negative then we can replace x, y, z with − x, − y, − z to obtain a solution in N. If two of them are negative, it must be x and z or y and z. If x, z are negative and y is positive, then we can rearrange to get (− z) n + y n = (− x) n resulting in a solution in N; the other case is dealt with analogously.

Now if just one is negative, it must be x or y. If x is negative, and y and z are positive, then it can be rearranged to get (− x) n + z n = y n again resulting in a solution in N; if y is negative, the result follows symmetrically.

Thus in all cases a nontrivial solution in Z would also mean a solution exists in N, the original formulation of the problem. Equivalent statement 2: x n + y n = z n, where integer n ≥ 3, has no non-trivial solutions x, y, z ∈ Q. This is because the exponent of x, y and z are equal (to n), so if there is a solution in Q then it can be multiplied through by an appropriate common denominator to get a solution in Z, and hence in N.

Equivalent statement 3: x n + y n = 1, where integer n ≥ 3, has no non-trivial solutions x, y ∈ Q. A non-trivial solution a, b, c ∈ Z to x n + y n = z n yields the non-trivial solution a / c, b / c ∈ Q for v n + w n = 1.

Conversely, a solution a / b, c / d ∈ Q to v n + w n = 1 yields the non-trivial solution ad, cb, bd for x n + y n = z n. This last formulation is particularly fruitful, because it reduces the problem from a problem about surfaces in three dimensions to a problem about curves in two dimensions. Furthermore, it allows working over the field Q, rather than over the ring Z; exhibit more structure than, which allows for deeper analysis of their elements.

Equivalent statement 4 – connection to elliptic curves: If a, b, c is a non-trivial solution to x p + y p = z p, p odd prime, then y 2 = x( x − a p)( x + b p) will be an. Examining this elliptic curve with shows that it does not have a. However, the proof by Andrew Wiles proves that any equation of the form y 2 = x( x − a n)( x + b n) does have a modular form. Any non-trivial solution to x p + y p = z p (with p an odd prime) would therefore create a, which in turn proves that no non-trivial solutions exist. In other words, any solution that could contradict Fermat's Last Theorem could also be used to contradict the Modularity Theorem. So if the modularity theorem were found to be true, then it would follow that no contradiction to Fermat's Last Theorem could exist either. As described above, the discovery of this equivalent statement was crucial to the eventual solution of Fermat's Last Theorem, as it provided a means by which it could be 'attacked' for all numbers at once.

Mathematical history Pythagoras and Diophantus Pythagorean triples. Main article: Around 1955, Japanese mathematicians and observed a possible link between two apparently completely distinct branches of mathematics, and. The resulting (at the time known as the Taniyama–Shimura conjecture) states that every elliptic curve is, meaning that it can be associated with a unique.

The link was initially dismissed as unlikely or highly speculative, but was taken more seriously when number theorist found evidence supporting it, though not proving it; as a result the conjecture was often known as the Taniyama–Shimura–Weil conjecture. It became a part of the, a list of important conjectures needing proof or disproof.: 211–215 The strategy that ultimately led to a successful proof of Fermat's Last Theorem arose from the 'astounding': 211, proposed around 1955—which many mathematicians believed would be near to impossible to prove,: 223 and was linked in the 1980s by, and to Fermat's equation.

By accomplishing a partial proof of this conjecture in 1994, ultimately succeeded in proving Fermat's Last Theorem, as well as leading the way to a full proof by others of what is now the. Even after gaining serious attention, the conjecture was seen by contemporary mathematicians as extraordinarily difficult or perhaps inaccessible to proof.: 203–205, 223, 226 For example, Wiles's doctoral supervisor states that it seemed 'impossible to actually prove',: 226 and Ken Ribet considered himself 'one of the vast majority of people who believed it was completely inaccessible', adding that 'Andrew Wiles was probably one of the few people on earth who had the audacity to dream that you can actually go and prove it.' Darkstorm vip keygen photoshop.

: 223 Ribet's theorem for Frey curves. Main articles: and In 1984, noted a link between Fermat's equation and the modularity theorem, then still a conjecture. If Fermat's equation had any solution ( a, b, c) for exponent p  2, then it could be shown that the semi-stable (now known as a ) y 2 = x ( x − a p)( x + b p) would have such unusual properties that it was unlikely to be modular. This would conflict with the modularity theorem, which asserted that all elliptic curves are modular.

As such, Frey observed that a proof of the Taniyama–Shimura–Weil conjecture might also simultaneously prove Fermat's Last Theorem. By, a disproof or refutation of Fermat's Last Theorem would disprove the Taniyama–Shimura–Weil conjecture. In plain English, Frey had shown that, if this intuition about his equation was correct, then any set of 4 numbers (a, b, c, n) capable of disproving Fermat's Last Theorem, could also be used to disprove the Taniyama–Shimura–Weil conjecture.

Therefore if the latter were true, the former could not be disproven, and would also have to be true. Following this strategy, a proof of Fermat's Last Theorem required two steps. First, it was necessary to prove the modularity theorem – or at least to prove it for the types of elliptical curves that included Frey's equation (known as ).

This was widely believed inaccessible to proof by contemporary mathematicians.: 203–205, 223, 226 Second, it was necessary to show that Frey's intuition was correct: that if an elliptic curve were constructed in this way, using a set of numbers that were a solution of Fermat's equation, the resulting elliptic curve could not be modular. Frey showed that this was plausible but did not go as far as giving a full proof. The missing piece (the so-called ', now known as ) was identified by who also gave an almost-complete proof and the link suggested by Frey was finally proved in 1986. Following Frey, Serre and Ribet's work, this was where matters stood:. Fermat's Last Theorem needed to be proven for all exponents n that were prime numbers.

The modularity theorem – if proved for semi-stable elliptic curves – would mean that all semistable elliptic curves must be modular. Ribet's theorem showed that any solution to Fermat's equation for a prime number could be used to create a semistable elliptic curve that could not be modular;. The only way that both of these statements could be true, was if no solutions existed to Fermat's equation (because then no such curve could be created), which was what Fermat's Last Theorem said.

As Ribet's Theorem was already proved, this meant that a proof of the Modularity Theorem would automatically prove Fermat's Last theorem was true as well. Wiles's general proof.

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Retrieved 5 August 2004. The title of one edition of the PBS television series NOVA, discusses Andrew Wiles's effort to prove Fermat's Last Theorem. Simon Singh and John Lynch's film tells the story of Andrew Wiles.