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Playing with equation in integers

Source: 2019 Belarus Team Selection Test 1.3

September 2, 2019
number theoryDiophantine equationPerfect Powers

Problem Statement

Given the equation ab⋅bc=ca a^b\cdot b^c=c^a in positive integers aa, bb, and cc. (i) Prove that any prime divisor of aa divides bb as well. (ii) Solve the equation under the assumption b≥ab\ge a. (iii) Prove that the equation has infinitely many solutions.
(I. Voronovich)
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