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An easy diophantine-like equation: find the exponent of 2

Source: Balkan MO 2001, problem 1

April 24, 2006
modular arithmeticnumber theory unsolvednumber theory

Problem Statement

Let a,b,na,b,n be positive integers such that 2n−1=ab2^n - 1 =ab. Let k∈Nk \in \mathbb N such that ab+a−b−1≡0(mod2k)ab+a-b-1 \equiv 0 \pmod {2^k} and ab+a−b−1≠0(mod2k+1)ab+a-b-1 \neq 0 \pmod {2^{k+1}}. Prove that kk is even.
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