MHB Square Number Pairs from 1-50: Counting Rules

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The discussion focuses on identifying integer pairs (x, y) from the range of 1 to 50, where at least one integer is a square number and their sum is also a square number. The mathematical relationship is defined by the equation $x^2 + y = z^2$, leading to the factorization $y = (z - x)(z + x)$. The author corrects an earlier sign error in the equations, clarifying that $y$ should be expressed as $y = z^2 - x^2$. This correction refines the approach to finding valid pairs and emphasizes the importance of accurate mathematical representation.

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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 ?
 
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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= (n-m)/2, just the opposite of what I had before.
 
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