Fermat’s Last Theorem - Elliptic Curves: Number Theory and Cryptography - page 442

Chapter 15

Fermat's Last Theorem

15.1 Overview

Around 1637, Fermat wrote in the margin of his copy of Diophantus's work
that, when n ≥ 3,

a n + b n = c n ,

c = 0 (15.1)

has no solution in integers a, b, c . This has become known as Fermat's Last
Theorem . Note that it suces to consider only the cases where n =4and
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 B n by the power series

− 1 = ∞

B n t n

t

n ! . e t

n =1

For example,

B 2 = 1

1
30 ,

691
2730 .

6 ,

B 4 = − ..., B 12 = −

Let be an odd prime. If does not divide the numerator of any of the
Bernoulli numbers

B 2 ,B 4 , ..., B − 3

then (15.1) has no solutions for n = . This criterion allowed Kummer to
prove Fermat's Last Theorem for all prime exponents less than 100, except
for =37 , 59 , 67. For example, 37 divides the numerator of the 32nd Bernoulli
number, so this criterion does not apply. Using more refined criteria, based on
the knowledge of which Bernoulli numbers are divisible by these exceptional
, Kummer was able to prove Fermat's Last Theorem for the three remaining