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if k so that abc = k(a+b+c) then a^3+b^3+c^3 not prime, in positive integers

Source: Spanish Mathematical Olympiad 1985 P4

August 2, 2018

number theorypositive integerprimeSum of powers

Problem Statement

Prove that for each positive integer

k

there exists a triple

(a,b,c)

of positive integers such that

abc = k(a+b+c)

. In all such cases prove that

a^3+b^3+c^3

is not a prime.