23044Re: Pythagorean Sets of Consecutive Primes
- Sep 1, 2011--- In firstname.lastname@example.org,
"Jens Kruse Andersen" <jens.k.a@...> wrote:
> If (a, b, c) and (b, c, d) are Pythagorean triples then is a=d?
Proof: Suppose the converse. Then we have 4 possibilities:
1) a^2 = b^2 + c^2, d^2 = b^2 - c^2
2) a^2 = b^2 + c^2, d^2 = c^2 - b^2
3) a^2 = b^2 - c^2, d^2 = b^2 + c^2
4) a^2 = c^2 - b^2, d^2 = b^2 + c^2
Hence there exists a pair of coprime integers (x,y)
such that x^2+y^2 and x^2-y^2 are both odd squares.
But each odd square is congruent to 1 modulo 8.
Hence 2*x^2 = 2 mod 8 and x is even.
Hence both y^2 and -y^2 are congruent to 1 modulo 4,
which is absurd.
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