Question 82
A positive integer $a$ is called a *double number* if it has an even number of digits (in base 10) and its base 10 representation has the form $a = a_1a_2\cdots a_k a_1a_2\cdots a_k$ with $0 \le a_i \le 9$ for $1 \le i \le k$, and $a_1 \ne 0$. For example, 283283 is a double number. Determine whether or not there are infinitely many double numbers $a$ such that $a + 1$ is a square and $a + 1$ is not a power of 10.