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Some explicit results on the sum of a prime and an almost prime

Daniel R. Johnston, Valeriia V. Starichkova

TL;DR

This work delivers explicit, effective bounds for representing even integers $N$ as the sum of a prime and an almost-prime. It achieves an unconditional result with $K=395$ prime factors and, under the Generalised Riemann Hypothesis, improves to $K=31$, by combining explicit linear sieve methods with detailed control over primes in arithmetic progressions and Siegel-zero phenomena. The conditional analysis leverages GRH to tighten error terms via ExplChebotarev bounds and a GRH-enabled Bombieri–Vinogradov framework, while the unconditional argument extends previous explicit variants of Chen’s theorem and refines parameter optimizations. The results, alongside discussions of potential improvements and computational avenues, advance explicit Goldbach-type representations and illustrate how sieve techniques interact with zero-free regions to yield sharp, verifiable constants.

Abstract

Inspired by a classical result of Rényi, we prove that every even integer $N\geq 4$ can be written as the sum of a prime and a number with at most 395 prime factors. We also show, under assumption of the generalised Riemann hypothesis, that this result can be improved to 31 prime factors.

Some explicit results on the sum of a prime and an almost prime

TL;DR

This work delivers explicit, effective bounds for representing even integers as the sum of a prime and an almost-prime. It achieves an unconditional result with prime factors and, under the Generalised Riemann Hypothesis, improves to , by combining explicit linear sieve methods with detailed control over primes in arithmetic progressions and Siegel-zero phenomena. The conditional analysis leverages GRH to tighten error terms via ExplChebotarev bounds and a GRH-enabled Bombieri–Vinogradov framework, while the unconditional argument extends previous explicit variants of Chen’s theorem and refines parameter optimizations. The results, alongside discussions of potential improvements and computational avenues, advance explicit Goldbach-type representations and illustrate how sieve techniques interact with zero-free regions to yield sharp, verifiable constants.

Abstract

Inspired by a classical result of Rényi, we prove that every even integer can be written as the sum of a prime and a number with at most 395 prime factors. We also show, under assumption of the generalised Riemann hypothesis, that this result can be improved to 31 prime factors.
Paper Structure (14 sections, 22 theorems, 98 equations, 2 figures)

This paper contains 14 sections, 22 theorems, 98 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Notation and setup
  3. List of definitions
  4. Some preliminary results
  5. The unconditional result
  6. The conditional result
  7. Conditional bounds on $\pi(x;q,a)$ and $\theta(x;q,a)$
  8. An extension of Lemma \ref{['djshehzadlem']}
  9. A conditional lower bound for $S(A,P(z))$
  10. Possible improvements
  11. Bounds on primes in arithmetic progressions
  12. Explicit bounds on Siegel zeros
  13. Bounds on sums and products of primes
  14. Further exploration of sieve methods

Key Result

Theorem 1.1

There exists a natural number $K$ such that every even integer $N\geq 4$ can be written as the sum of a prime and a number with at most $K$ prime factors.

Figures (2)

  • Figure 1: The scheme of the proof of Lemma \ref{['Lemma42-BJV']}. Each node indicates which lemma we need to use from BJV22 and how these lemmas must be updated for our purposes. An arrow $A\to B$ between two nodes indicates that the lemma $A$ is required to prove the lemma $B$.
  • Figure 2: The scheme of the proof of Theorem \ref{['theo:S>1']} in case (b).

Theorems & Definitions (42)

  • Theorem 1.1: Renyi1948
  • Theorem 1.2: Chen's Theorem
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 2.1: Kadiri2005
  • Theorem 2.2: Bordignon_21 & BJV22
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • ...and 32 more