On integers which are representable as sums of large squares

TL;DR

This paper proves that the largest integer not representable as a linear combination of large squares grows asymptotically as O(n^2), confirming a conjecture, and explores questions about sums of four large squares.

Contribution

It establishes the asymptotic growth of the largest non-representable integer for sums of large squares, verifying a prior conjecture and raising new questions.

Findings

Largest non-representable integer is O(n^2)

Confirmed Dutch and Rickett's conjecture

Raised questions on sums of four large squares

Abstract

We prove that the greatest positive integer that is not expressible as a linear combination with integer coefficients of elements of the set $\{n^2,(n+1)^2,\ldots \}$ is asymptotically $O(n^2)$, verifying thus a conjecture of Dutch and Rickett. Furthermore we ask a question on the representation of integers as sum of four large squares.