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Chinese factorial equation

Source: China Team Selection Test 2004, Day 1, Problem 2

October 13, 2005
floor functionlogarithmsnumber theory

Problem Statement

Let u be a fixed positive integer. Prove that the equation n!=uα−uβn! = u^{\alpha} - u^{\beta} has a finite number of solutions (n,α,β).(n, \alpha, \beta).
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