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

TL;DR

This paper explores the representation of natural numbers using floor functions of quadratic and triangular numbers, conjecturing universal representation except for specific cases, and confirms some cases with proofs and conjectures for future research.

Contribution

The paper introduces new conjectures on representing natural numbers with floor functions of quadratic and triangular numbers, and proves some special cases.

Findings

Confirmed that all positive integers can be represented as x^2 + y^2 + ⌊z^2/5⌋ with y odd.

Proved that for m=5,6,15, the set of sums of floor functions of quadratic numbers covers all non-negative integers.

Posed several conjectures including representation of integers as x^4 - y^3 + z^2.

Abstract

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 $$\left\{x^2+y^2+\left\lfloor\frac{z^2}5\right\rfloor:\ x,y,z\in\mathbb Z\ \mbox{and}\ 2\nmid y\right\}=\{1,2,3,\ldots\}$$ and $$\left\{\left\lfloor\frac{x^2}m\right\rfloor+\left\lfloor\frac{y^2}m\right\rfloor+\left\lfloor\frac{z^2}m\right\rfloor:\ x,y,z\in\mathbb Z\right\} =\{0,1,2,\ldots\}\ \ \mbox{for}\ m=5,6,15.$$ We also pose several conjectures for further research; for example, we conjecture that any integer can be written as $x^4-y^3+z^2$, where $x$, $y$ and $z$ are positive integers.