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Rational Square Sum (Posted on 2015-05-07) Difficulty: 3 of 5
Can the sum of squares of two rational numbers equal 14?
If so, give an example.
If not, prove it.

No Solution Yet Submitted by K Sengupta    
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re: Fixing the prior proof (spoiler) | Comment 3 of 5 |
(In reply to Fixing the prior proof (spoiler) by Steve Herman)

Steve, I think my proof holds up although I could/should have been clearer.  

A=2m+1, B=2n+1

A^2 + B^2 = 4m^2 + 4m + 1 + 4n^2 + 4n + 1 = 14C^2

2m^2 + 2m + 2n^2 + 2n + 1 = 7C^2

2m(m+1) + 2n(n+1) + 1 = 7C^2

Each of the first two terms is divisible by 4 since each is twice the product of consecutive integers.

However, overall I prefer your proof.  It's shorter and more direct and avoids any odd/even by-ways.

  Posted by xdog on 2015-05-07 20:09:23
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