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Infinitely many pos. integers m such that m - f(m) = 1989

Source: IMO Longlist 1989, Problem 9

September 18, 2008
floor functionnumber theory unsolvednumber theory

Problem Statement

Let m m be a positive integer and define f(m) f(m) to be the number of factors of 2 2 in m! m! (that is, the greatest positive integer k k such that 2k∣m! 2^k|m!). Prove that there are infinitely many positive integers m m such that m \minus{} f(m) \equal{} 1989.
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