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Solve n*5^{n-n/p} = p! (p^2+1) + n for prime p and natural n

Source: 2022 Abelkonkurransen Finale, Problem 1b

March 12, 2024
number theoryDiophantine Equations

Problem Statement

Find all primes pp and positive integers nn satisfying n⋅5n−n/p=p!(p2+1)+n.n \cdot 5^{n-n/p} = p! (p^2+1) + n.
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