ABSTRACT
Around 1637, Fermat wrote in the margin of his copy of Diophantus’s work that, when n ≥ 3,
an + bn = cn, abc = 0 (15.1)
has no solution in integers a, b, c. This has become known as Fermat’s Last Theorem. Note that it suffices to consider only the cases where n = 4 and where n = is an odd prime (since any n ≥ 3 has either 4 or such an as a factor). The case n = 4 was proved by Fermat using his method of infinite descent (see Section 8.6). At least one unsuccessful attempt to prove the case n = 3 appears in Arab manuscripts in the 900s (see [34]). This case was settled by Euler (and possibly by Fermat). The first general result was due to Kummer in the 1840s: Define the Bernoulli numbers Bn by the power series
t
et − 1 = ∞∑
Bn tn
n! .
