Let $p$ be a prime. For any diagonal quadratic form $f$, we define $Q_p(f)$ to be the set of integers represented by $f$ with each variable of $f$ is zero or not divisible by $p$. Then it seems to be interesting to determine the set $Q_p(f)$ when $f$ is an universal diagonal quaternary quadratic form.

In this talk, we consider this problem for the case when $f(x,y,z,w)=x^2+y^2+z^2+5w^2$.