Restricted sums of four squares

Abstract

We refine Lagrange's four-square theorem in new ways by imposing some restrictions involving powers of two (including $1$). For example, we show that each $n=1,2,3,\dots$ can be written as $x^2+y^2+z^2+w^2$ $(x,y,z,w\in \mathbb{N}=\{0,1,2,\dots \})$ with $|x+y-z|\in \{4^k:k\in \mathbb{N}\}$ (or $|2x-y|\in \{4^k:k\in \mathbb{N}\}$, or $x+y-z\in \{\pm 8^k:k\in \mathbb{N}\}\cup \{0\}\subseteq \{t^3:t\in \mathbb{Z}\}$), and that we can write any positive integer as $x^2+y^2+z^2+w^2$ $(x,y,z,w\in \mathbb{Z})$ with $x+y+2z$ (or $x+2y+2z$) a power of four. We also prove that any $n\in \mathbb{N}$ can be written as $x^2+y^2+z^2+2w^2$ $(x,y,z,w\in \mathbb{Z})$ with $x+y+z+w$ a square (or a cube). In addition, we pose some open conjectures for further research; for example, we conjecture that any integer $n>1$ can be written as $a^2+b^2+3^c+5^d$ with $a,b,c,d\in \mathbb{N}$.

31 pages, final published version

Cited by in corpus (1)

• Sums of squares with restrictions involving primes