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(x^2 - c)(y^2 -c) = z^2 -c , (x^2 + c)(y^2 - c) = z^2 - c

Source: JBMO 2009 Shortlist N5

October 14, 2017
JBMOnumber theorysystem of equations

Problem Statement

Show that there are infinitely many positive integers cc, such that the following equations both have solutions in positive integers: (x2−c)(y2−c)=z2−c(x^2 - c)(y^2 -c) = z^2 -c and (x2+c)(y2−c)=z2−c(x^2 + c)(y^2 - c) = z^2 - c.
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