Small prime solutions to cubic Diophantine equations

TL;DR

This paper proves the existence of small prime solutions to certain cubic diagonal equations under specific conditions, using the Hardy-Littlewood Circle method, advancing understanding of prime solutions in cubic Diophantine equations.

Contribution

It establishes new bounds and conditions for the solvability of cubic diagonal equations in primes, extending previous quadratic results to cubic cases.

Findings

Prime solutions exist with bounds depending on coefficients and n

Solvability conditions depend on the signs of coefficients

Uses Hardy-Littlewood Circle method for proofs

Abstract

Let a1,..., a9 be non-zero integers and n any integer. Suppose that a1 + ... + a9 = n (mod 2) and (ai, aj) = 1 for 1 <= i < j <= 9. We will prove that (i) if not all of the aj's are of the same sign, then the cubic diagonal equation a1p1^3 + ... + a9p9^3 = n has prime solutions satisfying pj << n^{1/3} + max{|aj|}^{20+ e}; and (ii) if all aj are positive and n >> max{|aj|}^{61+e}, then the cubic diagonal equation a1p1^3 + ... + a9p9^3 = n is soluble in primes pj. This result is motivated from the analogous result for quadratic diagonal equations by S.K.K. Choi and J. Liu. To prove the results we will use the Hardy-Littlewood Circle method, which we will outline. Lastly, we will make a note on some possible generalizations to this particular problem.