Problem
ALG-B1-M04-P020 Many Square Differences of Values
A quadratic polynomial \(P(x)\) is such that for some integers \(a\ne b\), the difference \(P(a)-P(b)\) is a square of a positive integer. Prove that there are more than \(1000\) integer pairs \((c,d)\) for which \(P(c)-P(d)\) is also a square of a positive integer.
Write \(P(x)=ux^2+vx+w\) and take \(c=a+k\), \(d=b-k\).
Let \(P(a)-P(b)=N^2\). For \(P(x)=ux^2+vx+w\), \(P(a)-P(b)=(a-b)(u(a+b)+v)\).
Take \(c=a+k\), \(d=b-k\). Then \(c+d=a+b\), and \(c-d=a-b+2k\), so \(P(c)-P(d)=(a-b+2k)(u(a+b)+v)\).
If \(k=(a-b)m\), then \(P(c)-P(d)=(2m+1)N^2\). Choose \(m\) so that \(2m+1\) is a square of an odd number: \(2m+1=(2t+1)^2\). For \(t=1,2,\ldots,1001\), we get more than \(1000\) different pairs.
Architecture preserved: symmetric movement of the points keeps the sum of arguments fixed.