The sum of a prime power and an almost prime
Daniel R. Johnston, Simon N. Thomas
TL;DR
This work addresses representing large integers as N = p^k + η with p prime and η constrained to have few prime factors. It develops and implements a weighted Diamond–Halberstam–Richert sieve framework, integrating bounds on representations of N as sums of two kth powers and results on kth-power residues, alongside Bombieri–Vinogradov and Elliott–Halberstam assumptions. The main contributions are explicit bounds M(k) and tilde M(k) that guarantee such representations for all large N (with M(k) = 6k for even k and M(k) = 4k for odd k; conditional improvements M(k) = (2+ε)k or (1+ε)k under EH), plus a congruence-conditional variant strengthening Erdős–Rao-type results. The paper also offers conjectures on optimal values of M(k), supported by computations, and connects to broader conjectures such as Hardy–Littlewood’s Conjecture H. Overall, it advances the quantitative understanding of Goldbach-type representations in the sparse setting of prime powers plus almost primes and provides a versatile sieve toolkit for related problems.
Abstract
For any fixed $k\geq 2$, we prove that every sufficiently large integer can be expressed as the sum of a $k$th power of a prime and a number with at most $M(k)=6k$ prime factors. For sufficiently large $k$ we also show that one can take $M(k)=(2+\varepsilon)k$ for any $\varepsilon>0$, or $M(k)=(1+\varepsilon)k$ under the assumption of the Elliott--Halberstam conjecture. Moreover, we give a variant of this result which accounts for congruence conditions and strengthens a classical theorem of Erdős and Rao. The main tools we employ are the weighted sieve method of Diamond, Halberstam and Richert, bounds on the number of representations of an integer as the sum of two $k$th powers, and results on $k$th power residues. We also use some simple computations and arguments to conjecture an optimal value of $M(k)$, as well as a related variant of Hardy and Littlewood's Conjecture H.