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The Problem: In the Pythagorean triplets (a,b,c) when a < b then b can't be a prime number.

The Background: While searching the properties of Pythagorean triplets in web I saw quite a few listed, but didn't see the above one which I thought was true, because I had developed a proof.

The Request: As discussed many a times in this site I would request some alternate proofs (or counterexamples) before I share mine for a review.

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$ 5^2 + 12^2 = 13^2 $ – Will Jagy Aug 7 '12 at 3:31
@Will, $12$ isn't prime, right? Or did Grothendieck say otherwise? – J. M. Aug 7 '12 at 3:36
@J.M., good point, I misread. Let me think. – Will Jagy Aug 7 '12 at 3:40
up vote 13 down vote accepted

Hint $\rm\,\ a^2\! + p^2 = c^2\:\Rightarrow\: p^2 = (c\!-\!a)(c\!+\!a).\:$ Unique factorization $\:\Rightarrow \begin{eqnarray}\rm\:c\!-\!a &=&1\\ \rm c\!+\!a &=&\rm p^2\end{eqnarray}\:$ contra $\rm\,a<p$

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My answer was something along your line -only the end logic was different. Since we know that $a<b<c=a+1$ It shows that $a<b<a+1$ implying $b$,an integer, should lie between two consecutive integers which is a fallacy. – Barun Dasgupta Aug 7 '12 at 17:29
@Barun Yes, that's the the proof I hinted at (I wanted to leave something for the reader). – Bill Dubuque Aug 7 '12 at 18:00

We use "Euclid's" formula for generating the Pythagorean triples. The job can be done much more elegantly, with much less machinery. Please see Bill Dubuque's answer.

The Pythagorean triples all have shape $k(x^2-y^2)$, $k(2xy)$, $k(x^2+y^2)$. Here $x$ and $y$ are positive integers such that $x \gt y$ (and $x$ and $y$ are relatively prime, and of opposite parity, though these side facts do not matter for the proof). If a leg is to be prime we need $k=1$.

Certainly $2xy$ cannot be prime. And $x^2-y^2=(x-y)(x+y)$ cannot be prime unless $x=y+1$. So from now on we may assume that $x=y+1$.

So $x^2-y^2=(y+1)^2-y^2=2y+1$. But $2xy=2(y-1)(y)=2y^2-2$. We cannot have $2y+1\gt 2y^2-2$. So the leg $2y+1$ must be the smallest. In particular, the larger of the two legs cannot be prime.

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Thanks Andre for the explanation. – Barun Dasgupta Aug 7 '12 at 17:26

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