Elliptic Curves over Q - Elliptic Curves: Number Theory and Cryptography - page 231

PROPOSITION 8.26
Suppose x, y, z are relativelyprimepositive integers such that

x 2 + y 2 = z 2 .

Then oneof x, y is even. Suppose itis x .Thenthere exist positive integers
m, n su ch that

y = m 2 n 2 , z = m 2 + n 2 . x =2 mn, −

M oreover, gcd( m, n )=1 and m

≡ n (mod 2) .

This result is proved in most elementary number theory texts. Alternatively,
see Exercise 2.21.
Suppose now that there are nonzero integers a, b, c satisfying (8.12). We
may assume a, b, c are positive and relatively prime. Proposition 8.26 implies
we may assume that a is even and that there exist integers m, n with

a 2 =2 mn, b 2 = m 2 n 2 , c = m 2 + n 2 .

−

If n is odd, then m is even, which implies that b 2

≡− 1 (mod 4). This is
impossible, so n is even and m is odd. Write n =2 q for some integer q .We
then have

( a/ 2) 2 = mq.

Since gcd( m, n ) = 1, we also have gcd( m, q ) = 1. Since m, q are relatively
prime and their product is a square, it follows easily from looking at the prime
factorizations of m, q that both m and q must be squares:

m = t 2 , q = u 2

for some positive integers t, u . Therefore, we have

b 2 = m 2 − n 2 = t 4 − 4 u 4 .

This may be rewritten as

(2 u 2 ) 2 + b 2 = t 4 .

Since m is odd, t is odd. Since gcd( m, q ) = 1, we also have gcd( t, u )=1.
Therefore, gcd( t, 2 u 2 ) = 1. Proposition 8.26 implies that

2 u 2 =2 vw, b = v 2 w 2 , 2 = v 2 + w 2 −

with gcd( v, w ) = 1. Since the product vw is a square, it follows that both v
and w are squares:

v = r 2 , w = s 2 .

Therefore, t 2 = v 2 + w 2 becomes

t 2 = r 4 + s 4 .