Sums of eight fourth power of primes

TL;DR

This paper proves that a positive proportion of large integers can be expressed as the sum of eight fourth powers of primes using circle method and sieve techniques.

Contribution

It establishes a lower bound on the density of integers representable as sums of eight prime fourth powers, advancing understanding of additive prime power representations.

Findings

Proves at least 1/414.465 of large integers are sums of eight prime fourth powers.

Uses circle method and sieve techniques for the proof.

Provides quantitative density results for such representations.

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 $\frac{1}{414.465}$ .