Sum of Squares of Two Odd IntegersDate: 10/26/1999 at 13:20:07 From: Devanshi Subject: Sum of the squares of 2 odd integers Hi there, I was looking for a proof of the following: The sum of the squares of 2 odd integers cannot be a perfect square. without using: The square of an odd integer equals 8k+1 for some integer k. Thanks, Devanshi Date: 10/26/1999 at 14:13:07 From: Doctor Anthony Subject: Re: Sum of the squares of 2 odd integers Suppose a,b,c is a set of integers such that a^2 + b^2 = c^2 Suppose too that both a and b are odd. Let a = 2m+1, b = 2n+1 a^2 = 4m^2 + 4m + 1 b^2 = 4n^2 + 4n + 1 then a^2 + b^2 = 4(m^2+n^2) + 4(m+n) + 2 = 2[2(m^2+n^2) + 2(m+n) + 1] = 2 * odd number and (2 * odd number) cannot be a perfect square. So we could not have a^2 + b^2 = c^2 with both a and b odd. Thus the sum of two odd integers cannot be a perfect square. - Doctor Anthony, The Math Forum http://mathforum.org/dr.math/ |
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