On certain quaternary quadratic forms

Abstract

In this paper, we determine all the positive integers $a,b$ and $c$ such that every nonnegative integer can be represented as $$f_c^{a,b}(x,y,z,w)=a{x}^2+b{y}^2+c(z^2+zw+w^2)\text{with}x,y,z,w\in \mathbb{Z}.$$ Furthermore, we prove that $f_c^{a,b}$ can represent all the nonnegative integers if it represents $n=1,2,3,5,6,10.$