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x^2 + y^2 + z^2 + t^2 - N * x * y * z * t - N = 0

Source: Vietnam TST 1994 for the 35th IMO, problem 2

June 25, 2005
calculusintegrationnumber theory unsolvednumber theory

Problem Statement

Consider the equation x2+y2+z2+t2−N⋅x⋅y⋅z⋅t−N=0x^2 + y^2 + z^2 + t^2 - N \cdot x \cdot y \cdot z \cdot t - N = 0 where NN is a given positive integer. a) Prove that for an infinite number of values of NN, this equation has positive integral solutions (each such solution consists of four positive integers x,y,z,tx, y, z, t), b) Let N=4⋅k⋅(8⋅m+7)N = 4 \cdot k \cdot (8 \cdot m + 7) where k,mk,m are no-negative integers. Prove that the considered equation has no positive integral solutions.
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