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Prove that we can find a,b,c

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September 20, 2010
quadraticsnumber theoryprimeSum of SquaresIMO Shortlist

Problem Statement

Prove that for any positive integers x,y,zx, y, z with xyāˆ’z2=1xy-z^2 = 1 one can find non-negative integers a,b,c,da, b, c, d such that x=a2+b2,y=c2+d2,z=ac+bdx = a^2 + b^2, y = c^2 + d^2, z = ac + bd. Set z=(2q)!z = (2q)! to deduce that for any prime number p=4q+1p = 4q + 1, pp can be represented as the sum of squares of two integers.
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