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China 2010 quiz1 Problem 3

Source:

September 5, 2010
modular arithmeticnumber theory unsolvednumber theory

Problem Statement

Fine all positive integers m,n≥2m,n\geq 2, such that (1) m+1m+1 is a prime number of type 4k−14k-1; (2) there is a (positive) prime number pp and nonnegative integer aa, such that m2n−1−1m−1=mn+pa.\frac{m^{2^n-1}-1}{m-1}=m^n+p^a.
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