3

Part of 2011 Ukraine Team Selection Test

Problems(1)

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

Source: Ukraine TST 2011 p3

Given a positive integer

n> 2

. Prove that there exists a natural

K

such that for all integers

k \ge K

on the open interval

({{k} ^{n}}, \ {{(k + 1)} ^{n}})

n

different integers, the product of which is the

n

-th power of an integer.

number theorypower of numberPerfect power