Fermat’s Last Theorem - Elliptic Curves: Number Theory and Cryptography - page 443

exponents. Refinements of Kummer's ideas by Vandiver and others, plus the
advent of computers, yielded extensions of Kummer's results to many more
exponents. For example, in 1992, Buhler, Crandall, Ernvall, and Metsankyla
proved Fermat's Last Theorem for all exponents less than 4 10 6 .Howcould

one check so many cases without seeing a pattern that would lead to a full
proof? The reason is that these methods were a prime-by-prime check. For
each prime , the Bernoulli numbers were computed mod . For around 61%
of the primes, none of these Bernoulli numbers was divisible by ,soKum-
mer's initial criterion yielded the result. For the remaining 39% of the primes,
more refined criteria were used, based on the knowledge of which Bernoulli
numbers were divisible by .For up to 4 × 10 6 , these criteria su ced to prove
the theorem. But it was widely suspected that eventually there would be ex-
ceptions to these criteria, and hence more refinements would be needed. The
underlying problem with this approach was that it did not include any con-
ceptual reason for why Fermat's Last Theorem should be true. In particular,
there was no reason why there couldn't be a few random exceptions.
In 1986, the situation changed. Suppose that

×

a + b = c , c =0 . (15.2)

By removing common factors, we may assume that a, b, c are integers with
gcd( a, b, c ) = 1, and by rearranging a, b, c and changing signs if necessary, we
may assume that

b ≡ 0(mod , a ≡− 1(mod . (15.3)

Frey suggested that the elliptic curve

y 2 = x ( x − a )( x + b )

E Frey :

(this curve had also been considered by Hellegouarch) has such restrictive
properties that it cannot exist, and therefore there cannot be any solutions to
(15.2). As we'll outline below, subsequent work of Ribet and Wiles showed
that this is the case.
When ≥ 5, the elliptic curve E Frey has good or multiplicative reduction
(see Exercise 2.24) at all primes (in other words, there is no additive reduc-
tion). Such an elliptic curve is called semistable . The discriminant of the
cubic is the square of the product of the differences of the roots, namely
a (

b )( a + b ) 2 =( abc ) 2 −

(we have used (15.2)). Because of technicalities involving the prime 2 (related
to the restrictions in (15.3)), the discriminant needs to be modified at 2 to
yield what is known as the minimal discriminant

Δ=2 − 8 ( abc ) 2