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

Abstract

Let $a,b,c$ be positive integers. It is known that there are infinitely many positive integers not representated by $a{x}^2+b{y}^2+c{z}^2$ 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 $(a,b,c)\ne (1,1,1),(2,2,2)$, 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.

28 pages. Add Conjectures 5.12-5.14

References in corpus (1)

• On sums of primes and triangular numbers

Cited by in corpus (1)

• Universal sums of three quadratic polynomials