Finite sequences of integers expressible as sums of two squares

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.