NL-BXMO-TST 2019 Problem 1

Prove that for each positive integer $n$ there are at most two pairs $(a, b)$ of positive integers with following two properties: (i) $a^2 + b = n$ , (ii) $a+b$ is a power of two, i.e. there is an integer $k \ge 0$ such that $a+b = 2^k$ .