On universal sums $ax^2+by^2+f(z)$, $aT_x+bT_y+f(z)$ and $aT_x+by^2+f(z)$

Zhi-Wei Sun

Published 2015-02-09Version 1

For integer-valued polynomials $f_1(x),f_2(y),f_3(z)$, if any $n\in\mathbb N=\{0,1,2,...\}$ can be written as $f_1(x)+f_2(y)+f_3(z)$ with $x,y,z\in\mathbb N$ then we say that $f_1(x)+f_2(y)+f_3(z)$ is universal (over $\mathbb N$). In this paper we find all candidates of universal sums of the following three types: $$ax^2+by^2+f(z),\ aT_x+bT_y+f(z),\ aT_x+by^2+f(z),$$ where $T_x$ denotes $x(x+1)/2$ and $f(z)$ has the form $c\binom z2+dz$ with $c,d\in\{1,2,3,...\}$ and $d\nmid c$. We also show that some of the candidates (including $T_x+y^2+z(z+2k)$ for $k=1,2,3$, and $T_x+y^2+z(z+2k+1)/2$ for $k=1,...,7$) are indeed universal sums.