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Sum of Squares of Two Odd Integers

Date: 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.


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


   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   
Associated Topics:
High School Discrete Mathematics
High School Number Theory

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