(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Let a,b,c be integers such that a^{2}+b^{2}=c^{2}.

Is it always true that at least one of {a,b} is even?

2. Relevant equations

3. The attempt at a solution

I say yes, and I am going to try and prove it with a proof by contrdiction.

Suppose a,b,c be integers such that a^{2}+b^{2}=c^{2}and that a and b are odd. By definition of odd a=2m+1, and b=2n+1, for some m,n in Z. By substitution we get c^{2}= (2m+1)^{2}+(2n+1)^{2}. By simple arithmatic we get c^{2}=2(2m^{2}+2n^{2}+2m+2n+1).

c=[tex]\sqrt{2}[/tex][tex]\sqrt{2m^2+2n^2+2m+2n+1}[/tex].

Because the second square root has an odd number in it means that we can not pull out a 1/[tex]\sqrt{2}[/tex] to cancel out the [tex]\sqrt{2}[/tex]. This means that we will have an irrational answer, rather than an integer for all m,n in Z. Thus we have reached a contradiction.//

I was wondering if my logic is correct on this.

Thank you for your time.

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# Homework Help: Pythagorean triples

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