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When 11...1 22...2 5 is a perfect square

Source: German TST 2004, IMO ShortList 2003, number theory problem 4

May 19, 2004
modular arithmeticnumber theorydecimal representationPerfect SquareIMO Shortlist

Problem Statement

Let b b be an integer greater than 5 5. For each positive integer n n, consider the number x_n = \underbrace{11\cdots1}_{n \minus{} 1}\underbrace{22\cdots2}_{n}5, written in base b b.
Prove that the following condition holds if and only if b \equal{} 10: there exists a positive integer M M such that for any integer n n greater than M M, the number xn x_n is a perfect square.
Proposed by Laurentiu Panaitopol, Romania
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