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Fearless Symmetry by Robert Gross, Avner Ash

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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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