
#1
March 27th, 2020,
01:37
Two integers will be taken from 1 to 50, where at least one of them should be a square number and sum of them should also be a square number. How many different pair like this can be found? Will I count (9,16) and (16,9) as one ?

March 27th, 2020 01:37
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#2
March 27th, 2020,
13:13
So $x^2+ y= z^2$ for x, y, and z integers. That is the same as $x^2 z^2= (x z)(x+ z)= y$. Look at the ways to factor y: y= mn and the x z= m, x+ z= n. Adding those two equations, 2x= m+ n, x= (m+ n)/2. Subtracting, 2z= n m, z= (n m)/2.
added much later: I've noticed that I have a sign error: from $x^2+ y= z^2$, $y= z^2 x^2$, not $x^2 x^2$. So y= (z x)(z+ x). Taking y= mn, z x= m, z+ x= n so that 2z= n+m, z= (n+m)/2, 2x= n m so x= (nm)/2, just the opposite of what I had before.
Last edited by HallsofIvy; April 3rd, 2020 at 14:17.