FERMAT’S LAST THEOREM AND GENERALIZED FERMAT EQUATIONS 255

the problem. For the rest of this chapter we will assume that p, q,

and r are all prime and that p is odd.

You can come up with “stupid” solutions to generalized Fermat

equations if you allow x, y,andz to share a common prime factor.

11

So from now on, we will assume that x, y, and z share no common

prime factor. We also assume that x, y,andz are all nonzero,

because solutions where one or more of the variables equals 0 are

easy to understand. We call solutions satisfying these assumptions

nontrivial primitive solutions. Our general feeling is that, for most

p, q, and r, there should be no nontrivial primitive solutions. We are

very far from knowing how to prove this, but some partial results

are known, examples of which are described in the rest of this

chapter and the next one.

What Henri Darmon and Lo

¨

ıc Merel Did

Darmon and Merel proved that

x

p

+ y

p

= z

r

has no nontrivial primitive solutions if r = 2 and p ≥ 5orifr = 3

and p ≥ 3. (Notice that 1

3

+ 2

3

= 3

2

gives a solution for r = 2 and

p = 3.)

Their methods follow the same general pattern as those of Wiles,

but various complications arise that limit the exponents to those

stated. For example, if (a, b, c) is a nontrivial primitive solution,

of x

p

+ y

p

= z

r

, instead of the Frey curve, they consider the elliptic

curves

y

2

= x

3

+ 2cx

2

+ a

p

x

if r = 2 and

y

2

= (x + 2c)(c

r

(x

3

− 3c

2

x) − 2(a

p

− b

p

))

11

For example, let a and b be any positive integers, and let c = a

p

+ b

p

. Then x = ac,

y = bc, and z = c

p+1

2

solves x

p

+ y

p

= z

2

.

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