Proof of Fermat's Last Theorem for specific exponents

Let g represent the greatest common divisor of a, b, and c. Then (a, b, c) may be written as a = gx, b = gy, and c = gz where the three numbers (x, y, z) are pairwise coprime.

The addition, subtraction and multiplication of even and odd integers obey simple rules.

This unique factorization property is the basis on which much of number theory is built.

One consequence of this unique factorization property is that if a pth power of a number equals a product such as and if u and v are coprime (share no prime factors), then u and v are themselves the pth power of two other numbers, u = rp and v = sp.

This fact led to the failure of Lamé's 1847 general proof of Fermat's Last Theorem.

Only one mathematical proof by Fermat has survived, in which Fermat 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.

In turn, this is sufficient to prove Fermat's Last Theorem for the case n = 4, since the equation a4 + b4 = c4 can be written as c4 − b4 = (a2)2.

Alternative proofs of the case n = 4 were developed later[2] by Frénicle de Bessy,[3] Euler,[4] Kausler,[5] Barlow,[6] Legendre,[7] Schopis,[8] Terquem,[9] Bertrand,[10] Lebesgue,[11] Pepin,[12] Tafelmacher,[13] Hilbert,[14] Bendz,[15] Gambioli,[16] Kronecker,[17] Bang,[18] Sommer,[19] Bottari,[20] Rychlik,[21] Nutzhorn,[22] Carmichael,[23] Hancock,[24] Vrǎnceanu,[25] Grant and Perella,[26] Barbara,[27] and Dolan.

Fermat's proof demonstrates that no right triangle with integer sides can have an area that is a square.

The first step of Fermat's proof is to factor the left-hand side[30] Since x and y are coprime (this can be assumed because otherwise the factors could be cancelled), the greatest common divisor of x2 + y2 and x2 − y2 is either 2 (case A) or 1 (case B).

Since (u, v, x) form a primitive Pythagorean triple they can be expressed in terms of smaller integers d and e using Euclid's formula Since u = 2m2 = 2de, and since d and e are coprime, they must be squares themselves, d = g2 and e = h2.

[31] Euler sent a letter to Goldbach on 4 August 1753 in which claimed to have a proof of the case in which n = 3.

[32] Euler had a complete and pure elementary proof in 1760, but the result was not published.

[34][35][36][37] Independent proofs were published by several other mathematicians,[38] including Kausler,[5] Legendre,[7][39] Calzolari,[40] Lamé,[41] Tait,[42] Günther,[43] Gambioli,[16] Krey,[44] Rychlik,[21] Stockhaus,[45] Carmichael,[46] van der Corput,[47] Thue,[48] and Duarte.

[50] The proof assumes a solution (x, y, z) to the equation x3 + y3 + z3 = 0, where the three non-zero integers x, y, and z are pairwise coprime and not all positive.

This implies that three does not divide u and that the two factors are cubes of two smaller numbers, r and s Since u2 + 3v2 is odd, so is s. A crucial lemma shows that if s is odd and if it satisfies an equation s3 = u2 + 3v2, then it can be written in terms of two integers e and f so that u and v are coprime, so e and f must be coprime, too.

That implies that 3 divides u, and one may express u = 3w in terms of a smaller integer, w. Since u is divisible by 4, so is w; hence, w is also even.

Fermat's Last Theorem for n = 5 states that no three coprime integers x, y and z can satisfy the equation This was proven[51] neither independently nor collaboratively by Dirichlet and Legendre around 1825.

[32][52] Alternative proofs were developed[53] by Gauss,[54] Lebesgue,[55] Lamé,[56] Gambioli,[16][57] Werebrusow,[58] Rychlik,[59] van der Corput,[47] and Terjanian.

Case A for n = 5 can be proven immediately by Sophie Germain's theorem if the auxiliary prime θ = 11.

[65] Alternative proofs were developed by Théophile Pépin[66] and Edmond Maillet.

Proofs for n = 6 have been published by Kausler,[5] Thue,[68] Tafelmacher,[69] Lind,[70] Kapferer,[71] Swift,[72] and Breusch.