Sums of eight fourth power of primes

Abstract

For any sufficiently large $\ell$, suppose that $\ell$ can be expressed as $$\ell =p_1^4+p_2^4+p_3^4+\cdots +p_8^4,$$ where $p_1,p_2,p_3,\cdots ,p_8$ are primes.For such $\ell$, in this paper we will use circle method and sieves to prove that the proportion of $\ell$ in positive integers is at least $\frac{1}{414.465}$ .