Abstract: In this talk, we will study the small prime solutions of certain diophantine equations by using the Hardy-Littlewood (circle) method. In particular, we prove the following. If $b_1,\cdots ,b_5$ are non-zero integers, then the quadratic equation $b_1p_1^2+\cdots +b_5p_5^2=n$ has prime solutions satisfying $p_j \ll \sqrt{|n|}+\max\{|b_j|\}^{20+\epsilon}$. In contrast to the earlier works which treat the enlarged major arc by the Deuring-Heilbronn phenomenon about the Siegel zero. We will explain the possible existence of Siegel zero does not have special influence and hence the Deuring-Heilbronn phenomenon can be avoided. This observation enables us to get better results without numerical computations.