Question 4
Pierre and Clara play the following game. First Clara chooses an integer $c$. Pierre then chooses a prime $p\geq c$ and writes two integers $a$ and $b$ on the board. Clara may perform the following operation: she chooses one of the two integers on the board, say $n$, erases it, and replaces it with an integer $m$ such that $p$ divides
$$(n^2-m+1)(m^2-n+1).$$
Clara wins if, after finitely many such operations, she makes the two integers on the board equal. Otherwise she loses and Pierre wins.
Prove that Pierre can prevent Clara from winning.
$$(n^2-m+1)(m^2-n+1).$$
Clara wins if, after finitely many such operations, she makes the two integers on the board equal. Otherwise she loses and Pierre wins.
Prove that Pierre can prevent Clara from winning.