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Solve $2^m p^2 + 1 = q^5$

Source: Finland 2013, Problem 5

May 2, 2013
modular arithmeticquadraticssearchnumber theoryFinland

Problem Statement

Find all integer triples (m,p,q)(m,p,q) satisfying 2mp2+1=q52^mp^2+1=q^5 where m>0m>0 and both pp and qq are prime numbers.
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