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Number of solutions of two systems

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September 15, 2010
number theoryperfect cubePerfect SquarecountingAdditive Number TheoryIMO ShortlistIMO Longlist

Problem Statement

Let NN be the number of integral solutions of the equation x2−y2=z3−t3x^2 - y^2 = z^3 - t^3 satisfying the condition 0≤x,y,z,t≤1060 \leq x, y, z, t \leq 10^6, and let MM be the number of integral solutions of the equation x2−y2=z3−t3+1x^2 - y^2 = z^3 - t^3 + 1 satisfying the condition 0≤x,y,z,t≤1060 \leq x, y, z, t \leq 10^6. Prove that N>M.N >M.
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