View source for Fermat great theorem

''Fermat's famous theorem, Fermat's big theorem, Fermat's last theorem''

The assertion that for any natural number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038390/f0383901.png" /> the equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038390/f0383902.png" /> (the Fermat equation) has no solution in non-zero integers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038390/f0383903.png" />. It was stated by P. Fermat in about 1630 in the margins of his copy of the book Aritmetika [[#References|[1]]] by Diophantus as follows:  "It is impossible to partition a cube into two cubes, or a biquadrate into two biquadrates, and in general any power greater than the second into two powers with the same exponent" . And he then added:  "I have discovered a truly marvellous proof of this, which this margin is too narrow to contain" . A proof of the theorem for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038390/f0383904.png" /> was found in Fermat's papers. No general proof has so far been obtained (1984), despite the efforts of many mathematicians (both professional and amateur). An unhealthy interest in proving this theorem was stimulated at one time by a large international prize, which was abolished at the end of the First World War.

It has been conjectured that there is no proof of Fermat's last theorem at all.

For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038390/f0383905.png" /> the theorem was proved by L. Euler (1770), for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038390/f0383906.png" /> by P.G.L. Dirichlet and A. Legendre (1825), and for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038390/f0383907.png" /> by G. Lamé (1839) (see [[#References|[2]]]). It is sufficient to prove the theorem for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038390/f0383908.png" /> and for every prime exponent <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038390/f0383909.png" />, that is, it is enough to prove that the equation

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038390/f03839010.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>

has no solutions in non-zero relatively-prime integers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038390/f03839011.png" />. One can also assume that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038390/f03839012.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038390/f03839013.png" /> are relatively prime to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038390/f03839014.png" />. For a proof of Fermat's theorem one considers two cases: case 1 when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038390/f03839015.png" />, and case 2 when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038390/f03839016.png" />. The proof of the second case is more difficult and is usually carried out by the method of infinite descent. An important contribution to proving Fermat's last theorem was made by E. Kummer, who invented a fundamentally new method based on his development of the arithmetic theory of a [[Cyclotomic field|cyclotomic field]]. It makes use of the fact that in the field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038390/f03839017.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038390/f03839018.png" />, the left-hand side of equation (1) splits into linear factors <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038390/f03839019.png" />, which are <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038390/f03839020.png" />-th powers of ideal numbers (cf. [[Ideal number|Ideal number]]) in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038390/f03839021.png" /> in case 1 and differ from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038390/f03839022.png" />-th powers by a factor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038390/f03839023.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038390/f03839024.png" />, in case 2. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038390/f03839025.png" /> divides the numerators of the [[Bernoulli numbers|Bernoulli numbers]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038390/f03839026.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038390/f03839027.png" />), then by the regularity criterion <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038390/f03839028.png" /> does not divide the class number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038390/f03839029.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038390/f03839030.png" /> and these ideal numbers are principal. Kummer

proved Fermat's theorem in this case. It is not known whether the number of regular numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038390/f03839031.png" /> is infinite or finite (by Jensen's theorem the number of irregular prime numbers is infinite [[#References|[4]]]). Kummer [[#References|[5]]] proved the theorem for some irregular prime numbers and also established its validity for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038390/f03839032.png" />. In case 1 he showed that (1) implies the congruences

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which are valid for any permutation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038390/f03839035.png" />. Hence he obtained that if equation (1) has a solution in case 1, then for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038390/f03839036.png" />,

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038390/f03839037.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>

In case 2 Kummer proved Fermat's theorem under the following conditions: 1) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038390/f03839038.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038390/f03839039.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038390/f03839040.png" /> is the first factor of the class number of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038390/f03839041.png" /> (this is equivalent to the requirement that only one of the numerators of the numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038390/f03839042.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038390/f03839043.png" />, is divisible by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038390/f03839044.png" />); 2) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038390/f03839045.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038390/f03839046.png" />); and 3) there is an ideal modulo which the unit

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is not congruent to the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038390/f03839048.png" />-th power of an integer in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038390/f03839049.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038390/f03839050.png" /> is a primitive root modulo <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038390/f03839051.png" /> and

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038390/f03839052.png" /></td> </tr></table>

Kummer's method has been widely developed in several articles on Fermat's last theorem (see [[#References|[6]]], [[#References|[7]]]). It has been established that (2) holds if (1) does in case 1 for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038390/f03839053.png" />, 9, 11, 13, 15, 17, 19. Under the same conditions M. Krasner [[#References|[8]]] showed that there is a number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038390/f03839054.png" /> such that for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038390/f03839055.png" /> (2) is true for all numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038390/f03839056.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038390/f03839057.png" />.

H. Brückner [[#References|[9]]] showed that the amount of numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038390/f03839058.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038390/f03839059.png" />, with numerators divisible by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038390/f03839060.png" /> is greater than <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038390/f03839061.png" />. Suppose that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038390/f03839062.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038390/f03839063.png" />. P. Remorov [[#References|[10]]] showed that there are constants <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038390/f03839064.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038390/f03839065.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038390/f03839066.png" />, such that for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038390/f03839067.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038390/f03839068.png" />, case 1 of the Fermat theorem is true. M. Eichler [[#References|[11]]] established that case 1 is true for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038390/f03839069.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038390/f03839070.png" /> is the index of irregularity of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038390/f03839071.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038390/f03839072.png" />. H. Vandiver [[#References|[12]]] proved case 1 for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038390/f03839073.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038390/f03839074.png" /> is the second factor of the class number of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038390/f03839075.png" />. He obtained interesting results on case 2 in [[#References|[13]]] and [[#References|[6]]]. For example, he showed that the Fermat theorem is true under the following conditions: 1) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038390/f03839076.png" />; and 2) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038390/f03839077.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038390/f03839078.png" />), <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038390/f03839079.png" />. The following theorem is most important: Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038390/f03839080.png" /> be an irregular prime number and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038390/f03839081.png" /> be the indices of the Bernoulli numbers among <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038390/f03839082.png" /> with numerators divisible by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038390/f03839083.png" />; if none of the units <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038390/f03839084.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038390/f03839085.png" />) is congruent to the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038390/f03839086.png" />-th power of an integer in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038390/f03839087.png" /> modulo <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038390/f03839088.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038390/f03839089.png" /> is the prime ideal dividing a prime number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038390/f03839090.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038390/f03839091.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038390/f03839092.png" />), then Fermat's theorem is true. From this Vandiver [[#References|[14]]] obtained an effectively-verifiable criterion for irregular prime numbers by means of which the Fermat theorem has been proved on a computer for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038390/f03839093.png" /> (see [[#References|[15]]]).

There are various results on case 1 of Fermat's last theorem. As early as 1823 Legendre published a result of S. Germain: If there is a prime number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038390/f03839094.png" /> such that the congruence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038390/f03839095.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038390/f03839096.png" />) has no integer solutions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038390/f03839097.png" /> not divisible by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038390/f03839098.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038390/f03839099.png" /> is not a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038390/f038390100.png" />-th power residue modulo <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038390/f038390101.png" />, then case 1 of the Fermat theorem holds (see [[#References|[2]]]). Hence he showed that if at least one of the numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038390/f038390102.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038390/f038390103.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038390/f038390104.png" />), <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038390/f038390105.png" />, is prime, then case 1 holds. This proposition has been extended to all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038390/f038390106.png" />. A. Wieferich [[#References|[16]]] discovered the following criterion: If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038390/f038390107.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038390/f038390108.png" /> is the Fermat quotient, then case 1 is true. D. Mirimanoff [[#References|[17]]] proved this for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038390/f038390109.png" />. Subsequently, case 1 was established by a number of other authors for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038390/f038390110.png" /> for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038390/f038390111.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038390/f038390112.png" /> is any prime number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038390/f038390113.png" />. From this the first case of Fermat's theorem follows for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038390/f038390114.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038390/f038390115.png" /> contain only prime numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038390/f038390116.png" /> in their prime factorizations. Calculations on a computer showed [[#References|[18]]] that among the numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038390/f038390117.png" /> only two: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038390/f038390118.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038390/f038390119.png" /> satisfy the condition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038390/f038390120.png" />, but for these <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038390/f038390121.png" />. This proves case 1 for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038390/f038390122.png" />. P. Furtwängler [[#References|[19]]] gave fairly simple new proofs of the results of Wieferich and Mirimanoff based on Eisenstein's reciprocity law. He also proved that if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038390/f038390123.png" /> is a solution of (1) and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038390/f038390124.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038390/f038390125.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038390/f038390126.png" /> but <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038390/f038390127.png" />, or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038390/f038390128.png" /> but <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038390/f038390129.png" />, or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038390/f038390130.png" /> but <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038390/f038390131.png" />.

A great variety of other criteria are known for case 1 of the Fermat theorem. They are connected with the solvability of certain congruences or with the existence of prime numbers of a certain form. The equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038390/f038390132.png" /> is not valid if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038390/f038390133.png" /> divides neither <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038390/f038390134.png" /> nor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038390/f038390135.png" /> (see [[#References|[20]]]). It is impossible in practice to produce a counterexample to Fermat's last theorem. K. Inkeri [[#References|[21]]] showed that if the integers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038390/f038390136.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038390/f038390137.png" />, satisfy (1), then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038390/f038390138.png" />, and in case 1: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038390/f038390139.png" />.

Fermat's last theorem can be stated as follows: For every natural number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038390/f038390140.png" /> there are no rational points on the Fermat curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038390/f038390141.png" /> except the trivial ones, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038390/f038390142.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038390/f038390143.png" />. Rational points on the Fermat curve have been studied by methods of algebraic geometry. By these methods it has been proved (1983) that the number of rational points on the Fermat curve is finite in every case. This follows from the [[Mordell conjecture|Mordell conjecture]], which was proved by G. Faltings [[#References|[23]]]. D.R. Heath-Brown has shown, using the Mordell conjecture, that Fermat's last theorem holds for almost-all primes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038390/f038390144.png" />, cf. [[#References|[24]]]. Also, by methods of [[Analytic number theory|analytic number theory]], L.M. Adleman, Foury and Heath-Brown have shown that case 1 holds for infinitely many primes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038390/f038390145.png" />, cf. [[#References|[25]]].

One can look at Fermat's equation in algebraic integers, entire functions, matrices, etc. There is a generalization of Fermat's theorem for equations of the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038390/f038390146.png" />.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  Diophantus of Alexandria,   "Aritmetika and the book on polygonal numbers" , Moscow  (1974)  (In Russian; translated from Greek)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  H.M. Edwards,   "Fermat's last theorem. A genetic introduction to algebraic number theory" , Springer  (1977)</TD></TR><TR><TD valign="top">[3a]</TD> <TD valign="top">  E. Kummer,   "Bestimmung der Anzahl nicht äquivalenter Classen für die aus <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038390/f038390147.png" />ten Wurzeln der Einheit gebildeten complexen Zahlen"  ''J. Reine Angew. Math.'' , '''40'''  (1850)  pp. 93–116</TD></TR><TR><TD valign="top">[3b]</TD> <TD valign="top">  E. Kummer,   "Zwei besondere Untersuchungen über die Classen-Anzahl und über die Einheiten der aus <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038390/f038390148.png" />ten Wurzeln der Einheit gebildeten complexen Zahlen"  ''J. Reine Angew. Math.'' , '''40'''  (1850)  pp. 117–129</TD></TR><TR><TD valign="top">[3c]</TD> <TD valign="top">  E. Krummer,   "Algemeiner Beweis des Fermat'schen Satzes, dass die Gleichung <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038390/f038390149.png" /> durch ganze Zahlen unlösbar ist, für diejenige Potenz-Exponenten <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038390/f038390150.png" />, welche ungerade Primzahlen sind und in den Zählern der ersten <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038390/f038390151.png" /> Bernoullischen Zahlen als Faktoren nicht vorkommen"  ''J. Reine Angew. Math.'' , '''40'''  (1950)  pp. 130–138</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  Z.I. Borevich,   I.R. Shafarevich,   "Number theory" , Acad. Press  (1966)  (Translated from Russian)  (German translation: Birkhäuser, 1966)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  E. Kummer,   "Einige Sätze über die aus den Wurzeln der Gleichung <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038390/f038390152.png" /> gebildeten complexen Zahlen, für den Fall, dass die Klassenanzahl durch <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038390/f038390153.png" /> teilbar ist, nebst Anwendung derselben auf einen weiteren Beweis des letzten Fermat'schen Lehrsatzes"  ''Abh. Akad. Wiss. Berlin, Math. Kl.''  (1857)  pp. 41–74</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  H. Vandiver,   "Fermat's last theorem"  ''Amer. Math. Monthly'' , '''53'''  (1946)  pp. 555–578</TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top">  P. Ribenboim,   "Thirteen lectures on Fermat's last theorem" , Springer  (1979)</TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top">  M. Krasner,   "Sur le premier cas du théorème de Fermat"  ''C.R. Acad. Sci. Paris'' , '''199'''  (1934)  pp. 256–258</TD></TR><TR><TD valign="top">[9]</TD> <TD valign="top">  H. Brückner,   "Zum Beweis des ersten Falles der Fermatschen Vermutung für pseudo requlären Primzahlen <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038390/f038390154.png" />"  ''J. Reine Angew. Math.'' , '''253'''  (1972)  pp. 15–18</TD></TR><TR><TD valign="top">[10]</TD> <TD valign="top">  P.N. Remorov,   "On Kummer's theorem"  ''Uchen. Zap. Leningrad. Gosudarstv. Univ. Ser. Mat. Nauk'' , '''144''' :  23  (1952)  pp. 26–34  (In Russian)</TD></TR><TR><TD valign="top">[11]</TD> <TD valign="top">  M. Eichler,   "Eine Bemerkung zur Fermatschen Vermutung"  ''Acta Arith.'' , '''11'''  (1965)  pp. 129–131</TD></TR><TR><TD valign="top">[12]</TD> <TD valign="top">  H. Vandiver,   "Fermat's last theorem and the second factor in the cyclotomic class number"  ''Bull. Amer. Math. Soc.'' , '''40'''  (1934)  pp. 118–126</TD></TR><TR><TD valign="top">[13]</TD> <TD valign="top">  H. Vandiver,   "On Fermat's last theorem"  ''Trans. Amer. Math. Soc.'' , '''31'''  (1929)  pp. 613–642</TD></TR><TR><TD valign="top">[14]</TD> <TD valign="top">  H.S. Vandiver,   "Examination of methods of attack on the second case of Fermat's last theorem"  ''Proc. Nat. Acad. Sci. USA'' , '''40''' :  8  (1954)  pp. 732–735</TD></TR><TR><TD valign="top">[15]</TD> <TD valign="top">  S. Wagstaff,   "The irregular primes to 125.000"  ''Math. Comp.'' , '''32'''  (1978)  pp. 583–591</TD></TR><TR><TD valign="top">[16]</TD> <TD valign="top">  A. Wieferich,   "Zum letzter Fermatschen Theorem"  ''J. Reine Angew. Math.'' , '''136'''  (1909)  pp. 293–302</TD></TR><TR><TD valign="top">[17]</TD> <TD valign="top">  D. Mirimanoff,   "Zum letzter Fermatschen Theorem"  ''J. Reine Angew. Math.'' , '''139'''  (1911)  pp. 309–324</TD></TR><TR><TD valign="top">[18]</TD> <TD valign="top">  D.H. Lehmer,   "On Fermat's quotient, base 2"  ''Math. Comp.'' , '''36'''  (1981)  pp. 289–290</TD></TR><TR><TD valign="top">[19]</TD> <TD valign="top">  P. Furtwängler,   "Letzter Fermat'scher Satz und Eisenstein'sches Reziprozitätsprinzip"  ''Sitzungsber. Akad. Wiss. Wien Math.-Naturwiss. Kl. IIa'' , '''121'''  (1912)  pp. 589–592</TD></TR><TR><TD valign="top">[20]</TD> <TD valign="top">  G. Terjanian,   "Sur l'equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038390/f038390155.png" />"  ''C.R. Acad. Sci. Paris'' , '''A285-B285''' :  16  (1977)  pp. 973A-975A  (English abstract)</TD></TR><TR><TD valign="top">[21]</TD> <TD valign="top">  K. Inkeri,   "Abschätzungen für eventuelle Lösungen der Gleichung im Fermatschen problem"  ''Ann. Univ. Turku Ser. A'' , '''16''' :  1  (1953)  pp. 3–9</TD></TR><TR><TD valign="top">[22]</TD> <TD valign="top">  M.M. Postnikov,   "An introduction to algebraic number theory" , Moscow  (1982)  (In Russian)</TD></TR><TR><TD valign="top">[23]</TD> <TD valign="top">  G. Faltings,   "Endichkeitssätze für abelsche Varietäten über Zahlkörpern"  ''Invent. Math.'' , '''73'''  (1983)  pp. 349–366</TD></TR><TR><TD valign="top">[24]</TD> <TD valign="top">  D.R. Heath-Brown,   "Fermat's last theorem for  "almost all"  exponents"  ''Bull. London Math. Soc.'' , '''17'''  (1985)  pp. 15–16</TD></TR><TR><TD valign="top">[25]</TD> <TD valign="top">  L.M. Adleman,   D.R. Heath-Brown,   "The first case of Fermat's last theorem"  ''Invent. Math.'' , '''79'''  (1985)  pp. 409–416</TD></TR></table>



====Comments====
In fact, Heath-Brown and, independently, A. Granville, cf. [[#References|[a1]]], have proved that the density of the exponents <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038390/f038390156.png" /> for which Fermat's last theorem holds is one.

It is now (1988) known that Fermat's last theorem holds for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038390/f038390157.png" />, and that case 1 holds for all primes up to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038390/f038390158.png" />, cf. [[#References|[a2]]].

Recently (1987), K. Ribet, using ideas of G. Frey and J.-P. Serre, showed that Fermat's last theorem is implied by the Weil–Taniyama conjecture in the theory of elliptic curves (cf. [[Elliptic curve|Elliptic curve]]).

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  S. Wagon,   "Fermat's last theorem"  ''Math. Intelligencer'' , '''8''' :  1  (1986)  pp. 59–61</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  P. Ribenboim,   "Recent results about Fermat's last theorem"  ''Cand. Math. Bull.'' , '''20'''  (1977)  pp. 229–242</TD></TR></table>