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

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