Natural numbers represented by $\lfloor x^2/a\rfloor+\lfloor y^2/b\rfloor+\lfloor z^2/c\rfloor$

Zhi-Wei Sun

Published 2015-04-06, updated 2015-04-13Version 3

Let $a,b,c$ be positive integers. It is known that there are infinitely many positive integers not representated by $ax^2+by^2+cz^2$ with $x,y,z\in\mathbb Z$. In contrast, we conjecture that any natural number is represented by $\lfloor x^2/a\rfloor+\lfloor y^2/b\rfloor +\lfloor z^2/c\rfloor$ with $x,y,z\in\mathbb Z$ if $(a,b,c)\not=(1,1,1),(2,2,2)$, and that any natural number is represented by $\lfloor T_x/a\rfloor+\lfloor T_y/b\rfloor+\lfloor T_z/c\rfloor$ with $x,y,z\in\mathbb Z$, where $T_x$ denotes the triangular number $x(x+1)/2$. We confirm this general conjecture in some special cases; in particular, we prove that $\{x^2+y^2+\lfloor z^2/m\rfloor:\ x,y,z\in\mathbb Z\}=\{0,1,2,...\}$ for all $m\in\{2,3,4,5,6,8,9,21\}$. We also conjecture that for any real number $\alpha\in(0,1.5]$ with $\alpha\not=1$ each positive integer can be written as the sum of three elements of the set $\{x^2+\lfloor \alpha x\rfloor:\ x\in\mathbb Z\}$ one of which is odd.