Elementary Number Theory by Underwood Dudley

Author:Underwood Dudley
Language: eng
Format: epub, pdf
ISBN: 9780486134871
Publisher: Dover Publications
Published: 2012-07-17T16:00:00+00:00

Lemma 3. Suppose that a, b, c is a fundamental solution of x2 + y2 = z2, and suppose that a is even. Then there are positive integers m and n with m > n, (m, n) = 1 and m ≢ n (mod 2) such that

a = 2mn,

b = m2 - n2,

c = m2 + n2.

(Note that we lose no generality in assuming that a is even. Lemma 1 tells us that exactly one of a and b is even, so we may as well call the even member of the pair a, b by the name of a.)

Proof. Since a is even, a = 2r for some r. So, a2 = 4r2; from a2 = c2 - b2 follows

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