Let $M$ denote the set consisting of positive integers that can be represented as $(a + b)(a^2 + b^2)$, where $a, b$ are some distinct positive integers. Prove that for any positive integer $n > k$, the number $n^4 - k^4$ is the sum of some $n - k$ distinct elements of the set $M$. Proposed by Oleksii Masalitin