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A divisibility condition and a sum being prime

Source: Baltic Way 2018, Problem 20

November 6, 2018
number theoryprime numbers

Problem Statement

Find all the triples of positive integers (a,b,c)(a,b,c) for which the number (a+b)4c+(b+c)4a+(c+a)4b\frac{(a+b)^4}{c}+\frac{(b+c)^4}{a}+\frac{(c+a)^4}{b} is an integer and a+b+ca+b+c is a prime.
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