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On a conjecture on shifted primes with large prime factors, II

Yuchen Ding

TL;DR

This paper addresses the distribution of shifted primes with large prime factors by studying the counting function $T_c(x)$, the number of primes $p\le x$ with $P^+(p-1)\ge p^c$. Employing a sieve-based approach, including a reduction to a two-variable sieve bound and Banks–Shparlinski-type estimates, the author proves a quantitative upper bound on the limsup: for $8/9\le c<1$, $\limsup_{x\to\infty} T_c(x)/\pi(x) \le 8\left(\frac{1}{c}-1\right)$. This yields an unconditional Erdős-type bound and clarifies the relationship to Chen–Chen conjectures, while also offering conditional refinements under the Elliott–Halberstam conjecture. The methods combine a sieve framework with Brun–Titchmarsh-type inequalities and the Dickman-related asymptotics to bound shifted-prime counts in terms of $x/\log x$, signaling where potential improvements in constants may appear under stronger hypotheses.

Abstract

Let $\mathcal{P}$ be the set of primes and $π(x)$ the number of primes not exceeding $x$. Let also $P^+(n)$ be the largest prime factor of $n$ with convention $P^+(1)=1$ and $$ T_c(x)=\#\left\{p\le x:p\in \mathcal{P},P^+(p-1)\ge p^c\right\}. $$ Motivated by a 2017 conjecture of Chen and Chen, we show that for any $8/9\le c<1$ $$ \limsup_{x\rightarrow\infty}T_c(x)/π(x)\le 8(1/c-1), $$ which clearly means that $$ \limsup_{x\rightarrow\infty}T_c(x)/π(x)\rightarrow 0, \quad \text{as}~c\rightarrow1. $$

On a conjecture on shifted primes with large prime factors, II

TL;DR

This paper addresses the distribution of shifted primes with large prime factors by studying the counting function , the number of primes with . Employing a sieve-based approach, including a reduction to a two-variable sieve bound and Banks–Shparlinski-type estimates, the author proves a quantitative upper bound on the limsup: for , . This yields an unconditional Erdős-type bound and clarifies the relationship to Chen–Chen conjectures, while also offering conditional refinements under the Elliott–Halberstam conjecture. The methods combine a sieve framework with Brun–Titchmarsh-type inequalities and the Dickman-related asymptotics to bound shifted-prime counts in terms of , signaling where potential improvements in constants may appear under stronger hypotheses.

Abstract

Let be the set of primes and the number of primes not exceeding . Let also be the largest prime factor of with convention and Motivated by a 2017 conjecture of Chen and Chen, we show that for any which clearly means that
Paper Structure (3 sections, 6 theorems, 35 equations)

This paper contains 3 sections, 6 theorems, 35 equations.

Table of Contents

  1. Introduction
  2. Proofs
  3. Remarks

Key Result

Proposition 1

Wu There exist two functions $K_2(\theta)>K_1(\theta)>0$, defined on the interval $(0,17/32)$ such that for each fixed real $A>0$, and all sufficiently large $Q=x^\theta$, the inequalities hold for all integers $m\in(Q,2Q]$ with at most $O\left(Q(\log Q)^{-A}\right)$ exceptions, where the implied constant depends only on $A$ and $\theta$. Moreover, for any fixed $\varepsilon>0$, these functions c

Theorems & Definitions (7)

  • Proposition 1
  • Theorem 1: Erdős
  • Theorem 2
  • Corollary 1
  • Lemma 1
  • Lemma 2
  • proof : Proof of Theorem \ref{['thm1']}