Question 23
Let b be an integer greater than 5. For each positive integer n, consider the number
$$ x_{n}=\underbrace{11\cdots1}_{n-1}\underbrace{22\cdots2}_{n}5, $$
written in base b.
Prove that the following condition holds if and only if b = 10:
there exists a positive integer M such that for any integer n greater than M, the number $ x_{n} $ is a perfect square.
$$ x_{n}=\underbrace{11\cdots1}_{n-1}\underbrace{22\cdots2}_{n}5, $$
written in base b.
Prove that the following condition holds if and only if b = 10:
there exists a positive integer M such that for any integer n greater than M, the number $ x_{n} $ is a perfect square.