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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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Solution Fixing the prior proof (spoiler) | Comment 2 of 5 |
xdog:

Your proof does not work mod 4.
If A,B are both odd, then 14C^2 = 2 mod 4.
It does not follow, however, that 7C^2 = 1 mod 4.  7C^2 could also = 3 mod 4, because 1*2 = 3*2 = 2 mod 4.
There is nothing to stop A, B and C from all being 1 mod 4.

However, your approach does work mod 7.
If A and B are both multiples of 7, then C necessarily is also.
Divide both sides by 49 as many times as necessary to get 
to a similar equation where A or B is not a multiple of 7.

Then the 14C^2 mod 7 = 0.
But A^2 + B^2 cannot = 0 mod 7,
because a square is = 1, 2, 4, or 0 mod 7,
but we have reduced to an equation where A^2 and B^2 cannot both be 0 mod 7.

  Posted by Steve Herman on 2015-05-07 19:11:54
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