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n different integers in (k^n,(k+1)^n), product of which is n-th power of integer

Source: Ukraine TST 2011 p3

May 7, 2020
number theorypower of numberPerfect power

Problem Statement

Given a positive integer n>2 n> 2 . Prove that there exists a natural K K such that for all integers k≥K k \ge K on the open interval (kn, (k+1)n) ({{k} ^{n}}, \ {{(k + 1)} ^{n}}) there are nn different integers, the product of which is the nn-th power of an integer.
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