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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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Generalisation. | Comment 4 of 5 |

Let (a/x)^2+(b/y)^2=c, where the rational fractions cannot be further resolved. It's easily shown that a^2+b^2=cx^2, for some integer, x (see, e.g. xdogs' first post).

But if so, then x = (a^2+b^2)^(1/2)/c^(1/2), but since we are dealing with square roots, once the fraction is again fully resolved, this can be true only if c=a^2+b^2, i.e. x=1, or a^2+b^2=x^2, i.e. c=1.

In either case, it seems that c is the sum of rational squares iff it is the sum of integer squares, a non-trivial result.

Since 14 is not a sum of integer squares, the solution of the problem follows at once.

Edited on May 8, 2015, 10:25 am
  Posted by broll on 2015-05-08 09:15:06

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