The proposition was first conjectured by Pierre de Fermat around in the margin of a copy of Arithmetica ; Fermat added that he had a proof that was too large to fit in the margin. However, there were doubts that he had a correct proof because his claim was published by his son without his consent and after his death. The unsolved problem stimulated the development of algebraic number theory in the 19th century and the proof of the modularity theorem in the 20th century. It is among the most notable theorems in the history of mathematics and prior to its proof was in the Guinness Book of World Records as the "most difficult mathematical problem" in part because the theorem has the largest number of unsuccessful proofs.

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The proposition was first conjectured by Pierre de Fermat around in the margin of a copy of Arithmetica ; Fermat added that he had a proof that was too large to fit in the margin. However, there were doubts that he had a correct proof because his claim was published by his son without his consent and after his death.

The unsolved problem stimulated the development of algebraic number theory in the 19th century and the proof of the modularity theorem in the 20th century. It is among the most notable theorems in the history of mathematics and prior to its proof was in the Guinness Book of World Records as the "most difficult mathematical problem" in part because the theorem has the largest number of unsuccessful proofs.

Although he claimed to have a general proof of his conjecture, Fermat left no details of his proof, and no proof by him has ever been found. This claim, which came to be known as Fermat's Last Theorem , stood unsolved for the next 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 number theory , and over time Fermat's Last Theorem gained prominence as an unsolved problem in mathematics.

In the midth century, Ernst Kummer extended this and proved the theorem for all regular primes , 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.

Separately, around , Japanese mathematicians Goro Shimura and Yutaka Taniyama suspected a link might exist between elliptic curves and modular forms , two completely different areas of mathematics. Known at the time as the Taniyama—Shimura conjecture eventually as the modularity theorem , 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 , Gerhard Frey 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 by Ken Ribet , building on a partial proof by Jean-Pierre Serre , who proved all but one part known as the "epsilon conjecture" see: Ribet's Theorem and Frey curve. The connection is described below : any solution that could contradict Fermat's Last Theorem could also be used to contradict the Taniyama—Shimura conjecture.

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 and widely considered to be "completely inaccessible" to proof at the time, [3] 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.

Unlike Fermat's Last Theorem, the Taniyama—Shimura conjecture was a major active research area and viewed as more within reach of contemporary mathematics. On hearing that Ribet had proven Frey's link to be correct, English mathematician Andrew Wiles , 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 , after six years of working secretly on the problem, Wiles succeeded in proving 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 peer review and required a further year and collaboration with a past student, Richard Taylor , to resolve.

As a result, the final proof in was accompanied by a 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 modularity theorem, were subsequently proved by other mathematicians, who built on Wiles's work between and 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, A solution where all three are non-zero will be called a non-trivial solution. Most popular treatments of the subject state it this way. In contrast, almost all mathematics textbooks [ which? The equivalence is clear if n is even. If two of them are negative, it must be x and z or y and z. Now if just one is negative, it must be x or y. Thus in all cases a nontrivial solution in Z would also mean a solution exists in N , the original formulation of the problem.

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. 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 ; fields exhibit more structure than rings , which allows for deeper analysis of their elements.

Examining this elliptic curve with Ribet's theorem shows that it does not have a modular form. 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. In ancient times it was known that a triangle whose sides were in the ratio would have a right angle as one of its angles.

This was used in construction and later in early geometry. This is now known as the Pythagorean theorem , and a triple of numbers that meets this condition is called a Pythagorean triple — both are named after the ancient Greek Pythagoras.

Examples include 3, 4, 5 and 5, 12, There are infinitely many such triples, [12] and methods for generating such triples have been studied in many cultures, beginning with the Babylonians [13] and later ancient Greek , Chinese , and Indian mathematicians. A typical Diophantine problem is to find two integers x and y such that their sum, and the sum of their squares, equal two given numbers A and B , respectively:. Diophantus's major work is the Arithmetica , of which only a portion has survived.

Diophantine equations have been studied for thousands of years. Problem II. Around , Fermat wrote his Last Theorem in the margin of his copy of the Arithmetica next to Diophantus's sum-of-squares problem : [22]. It is not known whether Fermat had actually found a valid proof for all exponents n , but it appears unlikely.

Van der Poorten [30] suggests that while the absence of a proof is insignificant, the lack of challenges means Fermat realised he did not have a proof; he quotes Weil [31] as saying Fermat must have briefly deluded himself with an irretrievable idea.

Taylor and Wiles's proof relies on 20th-century techniques. Only one relevant proof by Fermat has survived, in which he uses the technique of infinite descent to show that the area of a right triangle with integer sides can never equal the square of an integer.

The general equation. Nevertheless, the reasoning of these even-exponent proofs differs from their odd-exponent counterparts. All proofs for specific exponents used Fermat's technique of infinite descent , [ citation needed ] either in its original form, or in the form of descent on elliptic curves or abelian varieties.

The details and auxiliary arguments, however, were often ad hoc and tied to the individual exponent under consideration. In the early 19th century, Sophie Germain developed several novel approaches to prove Fermat's Last Theorem for all exponents.

Although she developed many techniques for establishing the non-consecutivity condition, she did not succeed in her strategic goal. His proof failed, however, because it assumed incorrectly that such complex numbers can be factored uniquely into primes, similar to integers. This gap was pointed out immediately by Joseph Liouville , who later read a paper that demonstrated this failure of unique factorisation, written by Ernst Kummer.

Kummer set himself the task of determining whether the cyclotomic field could be generalized to include new prime numbers such that unique factorisation was restored. He succeeded in that task by developing the ideal numbers. Harold Edwards says the belief that Kummer was mainly interested in Fermat's Last Theorem "is surely mistaken". In the s, Louis Mordell posed a conjecture that implied that Fermat's equation has at most a finite number of nontrivial primitive integer solutions, if the exponent n is greater than two.

In the latter half of the 20th century, computational methods were used to extend Kummer's approach to the irregular primes. However despite these efforts and their results, no proof existed of Fermat's Last Theorem. Proofs of individual exponents by their nature could never prove the general case: even if all exponents were verified up to an extremely large number X, a higher exponent beyond X might still exist for which the claim was not true.

This had been the case with some other past conjectures, and it could not be ruled out in this conjecture. The strategy that ultimately led to a successful proof of Fermat's Last Theorem arose from the "astounding" [] : Taniyama—Shimura—Weil conjecture , proposed around —which many mathematicians believed would be near to impossible to prove, [] : and was linked in the s by Gerhard Frey , Jean-Pierre Serre and Ken Ribet to Fermat's equation.

By accomplishing a partial proof of this conjecture in , Andrew Wiles 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 modularity theorem. Around , Japanese mathematicians Goro Shimura and Yutaka Taniyama observed a possible link between two apparently completely distinct branches of mathematics, elliptic curves and modular forms.

The resulting modularity theorem at the time known as the Taniyama—Shimura conjecture states that every elliptic curve is modular , meaning that it can be associated with a unique modular form. Even after gaining serious attention, the conjecture was seen by contemporary mathematicians as extraordinarily difficult or perhaps inaccessible to proof.

In , Gerhard Frey noted a link between Fermat's equation and the modularity theorem, then still a 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 semistable elliptic curves.

This was widely believed inaccessible to proof by contemporary mathematicians. Frey showed that this was plausible but did not go as far as giving a full proof.

The missing piece the so-called " epsilon conjecture ", now known as Ribet's theorem was identified by Jean-Pierre Serre who also gave an almost-complete proof and the link suggested by Frey was finally proved in by Ken Ribet. Ribet's proof of the epsilon conjecture in accomplished the first of the two goals proposed by Frey.

Upon hearing of Ribet's success, Andrew Wiles , an English mathematician with a childhood fascination with Fermat's Last Theorem, and who had worked on elliptic curves, decided to commit himself to accomplishing the second half: proving a special case of the modularity theorem then known as the Taniyama—Shimura conjecture for semistable elliptic curves. Wiles worked on that task for six years in near-total secrecy, covering up his efforts by releasing prior work in small segments as separate papers and confiding only in his wife.

Since his work relied extensively on this approach, which was new to mathematics and to Wiles, in January he asked his Princeton colleague, Nick Katz , to help him check his reasoning for subtle errors. Their conclusion at the time was that the techniques Wiles used seemed to work correctly. By mid-May , Wiles felt able to tell his wife he thought he had solved the proof of Fermat's Last Theorem, [] : and by June he felt sufficiently confident to present his results in three lectures delivered on 21—23 June at the Isaac Newton Institute for Mathematical Sciences.

However, it became apparent during peer review that a critical point in the proof was incorrect. It contained an error in a bound on the order of a particular group. The error was caught by several mathematicians refereeing Wiles's manuscript including Katz in his role as reviewer , [] who alerted Wiles on 23 August The error would not have rendered his work worthless — each part of Wiles's work was highly significant and innovative by itself, as were the many developments and techniques he had created in the course of his work, and only one part was affected.

Wiles spent almost a year trying to repair his proof, initially by himself and then in collaboration with his former student Richard Taylor , without success. Mathematicians were beginning to pressure Wiles to disclose his work whether it was complete or not, so that the wider community could explore and use whatever he had managed to accomplish.

But instead of being fixed, the problem, which had originally seemed minor, now seemed very significant, far more serious, and less easy to resolve.

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## Fermat's Last Theorem

Fermaova poslednja teorema poznata i kao Fermaova velika teorema je jedna od najpoznatijih teorema u istoriji matematike. Ona tvrdi da:. Original latinski : "Cuius rei demonstrationem mirabilem sane detexi hanc marginis exiguitas non caperet. Teorema nije poslednja koju je Ferma dao, nego poslednja koja treba biti dokazana. Gerd Faltings je Ken Ribet je

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## Fermatov posljednji teorem

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## Sajmon Sing - Fermaova Poslednja Teorema, 1. Pitagora

Shelves: stem-shelf-research Before delving into the book itself, I thought Id start things off by introducing the problem its concerned with, just in case you arent already familiar with it. So, what exactly is Fermats Last Theorem? Well, basically, this is it: As you can see, the conjecture is quite easy to understand, and yet, believe it or not, it was so remarkably difficult to prove that it took over years to accomplish! The fact that Fermat teasingly? There are a handful of fairly simple proofs included in the appendices, but overall, the concepts under discussion are glossed over in a superficial manner, never examined in any kind of detail. In any case, while Singh did not pursue the actual mathematics in any real sense, he did positively excel at telling the story of an utterly fascinating struggle, one which spanned hundreds of years and ensnared countless brilliant, talented minds.

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