On the almost universality of $\lfloor x^2/a\rfloor+\lfloor y^2/b\rfloor+\lfloor z^2/c\rfloor$

Hai-Liang Wu, He-Xia Ni, Hao Pan

Published 2018-06-26Version 1

For each real number $r$, let $\lfloor r\rfloor$ denote the integral part of $r$. In this paper, with the help of congruence theta functions, we obtain the following results: (i) For each integer $m\ge3$, every sufficiently large integer $n$ can be written as $$\lfloor x^2/m\rfloor+\lfloor y^2/m\rfloor+\lfloor z^2/m\rfloor,$$ with $x,y,z\in\Z$. (ii) For any integers $a,b,c\ge5$, assume that $a,b,c$ are pairwisely coprime, then every sufficiently large integer $n$ can be written as $$\lfloor x^2/a\rfloor+\lfloor y^2/b\rfloor+\lfloor z^2/c\rfloor,$$ with $x,y,z\in\Z$.