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if abc = k^2 + 1 in N, then one of a - 1, b - 1, c - 1 is a composite

Source: INAMO Shortlist 2014 N2

July 9, 2019
number theorycomposite numberCompositeDiophantine equation

Problem Statement

Suppose that a,b,c,ka, b, c, k are natural numbers with a,b,c≥3a, b, c \ge 3 which fulfill the equation abc=k2+1abc = k^2 + 1. Show that at least one between a−1,b−1,c−1a - 1, b - 1, c -1 is composite number.
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