Finite sequences of integers expressible as sums of two squares (2310.13317v2)

Abstract: This paper is concerned with finite sequences of integers that may be written as sums of squares of two nonzero integers. We first find infinitely many integers $n$ such that $n, n+h$ and $n+k$ are all sums of two squares where $h$ and $k$ are two arbitrary integers, and as an immediate corollary obtain, in parametric terms, three consecutive integers that are sums of two squares. Similarly we obtain $n$ in parametric terms such that all the four integers $n, n+1, n+2, n+4$ are sums of two squares. We also find infinitely many integers $n$ such that all the five integers $n, n+1, n+2, n+4, n+5$ are sums of two squares, and finally, we find infinitely many arithmetic progressions, with common difference $4$, of five integers all of which are sums of two squares.