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p^{q-1}- q^{p-1}=4n^2 diophantine without solution for primes p \ne q

Source: Balkan MO Shortlist 2013 N5 BMO

March 9, 2020
number theoryDiophantine equationdiophantineprimes

Problem Statement

Prove that there do not exist distinct prime numbers pp and qq and a positive integer nn satisfying the equation pq−1−qp−1=4n2p^{q-1}- q^{p-1}=4n^2
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