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product of sum becoming exponents

Source: ARO 2008, Grade 10

June 12, 2008
number theory unsolvednumber theory

Problem Statement

For which integers n>1 n>1 do there exist natural numbers b1,b2,…,bn b_1,b_2,\ldots,b_n not all equal such that the number (b_1\plus{}k)(b_2\plus{}k)\cdots(b_n\plus{}k) is a power of an integer for each natural number k k? (The exponents may depend on k k, but must be greater than 1 1)
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