Fermat's Last Theorem - Proofs For Specific Exponents

Proofs For Specific Exponents

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. His proof is equivalent to demonstrating that the equation

has no primitive solutions in integers (no pairwise coprime solutions). In turn, this proves 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 by Frénicle de Bessy (1676), Leonhard Euler (1738), Kausler (1802), Peter Barlow (1811), Adrien-Marie Legendre (1830), Schopis (1825), Terquem (1846), Joseph Bertrand (1851), Victor Lebesgue (1853, 1859, 1862), Theophile Pepin (1883), Tafelmacher (1893), David Hilbert (1897), Bendz (1901), Gambioli (1901), Leopold Kronecker (1901), Bang (1905), Sommer (1907), Bottari (1908), Karel Rychlík (1910), Nutzhorn (1912), Robert Carmichael (1913), Hancock (1931), and Vrǎnceanu (1966).

For another proof for n=4 by infinite descent, see Infinite descent: Non-solvability of r2 + s4 = t4. For various proofs for n=4 by infinite descent, see Grant and Perella (1999), Barbara (2007), and Dolan (2011).

After Fermat proved the special case n = 4, the general proof for all n required only that the theorem be established for all odd prime exponents. In other words, it was necessary to prove only that the equation an + bn = cn has no integer solutions (a, b, c) when n is an odd prime number. This follows because a solution (a, b, c) for a given n is equivalent to a solution for all the factors of n. For illustration, let n be factored into d and e, n = de. The general equation

an + bn = cn

implies that (ad, bd, cd) is a solution for the exponent e

(ad)e + (bd)e = (cd)e.

Thus, to prove that Fermat's equation has no solutions for n > 2, it suffices to prove that it has no solutions for at least one prime factor of every n. All integers n > 2 contain a factor of 4, or an odd prime number, or both. Therefore, Fermat's Last Theorem can be proven for all n if it can be proven for n = 4 and for all odd primes (the only even prime number is the number 2) p.

In the two centuries following its conjecture (1637–1839), Fermat's Last Theorem was proven for three odd prime exponents p = 3, 5 and 7. The case p = 3 was first stated by Abu-Mahmud Khojandi (10th century), but his attempted proof of the theorem was incorrect. In 1770, Leonhard Euler gave a proof of p = 3, but his proof by infinite descent contained a major gap. However, since Euler himself had proven the lemma necessary to complete the proof in other work, he is generally credited with the first proof. Independent proofs were published by Kausler (1802), Legendre (1823, 1830), Calzolari (1855), Gabriel Lamé (1865), Peter Guthrie Tait (1872), Günther (1878), Gambioli (1901), Krey (1909), Rychlík (1910), Stockhaus (1910), Carmichael (1915), Johannes van der Corput (1915), Axel Thue (1917), and Duarte (1944). The case p = 5 was proven independently by Legendre and Peter Dirichlet around 1825. Alternative proofs were developed by Carl Friedrich Gauss (1875, posthumous), Lebesgue (1843), Lamé (1847), Gambioli (1901), Werebrusow (1905), Rychlík (1910), van der Corput (1915), and Guy Terjanian (1987). The case p = 7 was proven by Lamé in 1839. His rather complicated proof was simplified in 1840 by Lebesgue, and still simpler proofs were published by Angelo Genocchi in 1864, 1874 and 1876. Alternative proofs were developed by Théophile Pépin (1876) and Edmond Maillet (1897).

Fermat's Last Theorem has also been proven for the exponents n = 6, 10, and 14. Proofs for n = 6 have been published by Kausler, Thue, Tafelmacher, Lind, Kapferer, Swift, and Breusch. Similarly, Dirichlet and Terjanian each proved the case n = 14, while Kapferer and Breusch each proved the case n = 10. Strictly speaking, these proofs are unnecessary, since these cases follow from the proofs for n = 3, 5, and 7, respectively. Nevertheless, the reasoning of these even-exponent proofs differs from their odd-exponent counterparts. Dirichlet's proof for n = 14 was published in 1832, before Lamé's 1839 proof for n = 7.

Many proofs for specific exponents use Fermat's technique of infinite descent, which Fermat used to prove the case n = 4, but many do not. However, the details and auxiliary arguments are often ad hoc and tied to the individual exponent under consideration. Since they became ever more complicated as p increased, it seemed unlikely that the general case of Fermat's Last Theorem could be proven by building upon the proofs for individual exponents. Although some general results on Fermat's Last Theorem were published in the early 19th century by Niels Henrik Abel and Peter Barlow, the first significant work on the general theorem was done by Sophie Germain.