But each \((m,n)\) is distinct, so 16 solutions.

February 22, 2026

$But each \((m,n)\) is distinct, so 16 solutions.$

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However, consider symmetry: for example, \((m,n)\) and \((n,m)\) give:
So symmetric about x-axis. But both are distinct unless \(y = 0\).
But since 506 ≠ 0, no solution has \(y = 0\) (would require \(m = n\), \(m^2 = 506\), not square), so all solutions come in pairs \((x,y)\) and \((x,-y)\), except if \(y = 0\), which doesn’t occur.
But let’s test a small one: \(m = 1\), \(n = 506\):
\(a = 2\), \(b = 1012\), then \(x = (2+1012)/2 = 507\), \(y = (1012 - 2)/2 = 505\).
Check: \(507^2 - 505^2 = (507-505)(507+505) = 2 \cdot 1012 = 2024\), correct.

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