On Values Taken by the Largest Prime Factor of Shifted Primes

Abstract

Denote by $\mathbb{P}$ the set of all prime numbers and by $P(n)$ the largest prime factor of positive integer $n\geq 1$ with the convention $P(1)=1$. In this paper, we prove that, for each $\unicode[STIX]{x1D702}\in (\frac{32}{17},2.1426\cdots \,)$, there is a constant $c(\unicode[STIX]{x1D702})>1$ such that, for every fixed nonzero integer $a\in \mathbb{Z}^{\ast }$, the set $$\begin{eqnarray}\{p\in \mathbb{P}:p=P(q-a)\text{ for some prime }q\text{ with }p^{\unicode[STIX]{x1D702}}<q\leq c(\unicode[STIX]{x1D702})p^{\unicode[STIX]{x1D702}}\}\end{eqnarray}$$ has relative asymptotic density one in $\mathbb{P}$. This improves a similar result due to Banks and Shparlinski [‘On values taken by the largest prime factor of shifted primes’, J. Aust. Math. Soc.82 (2015), 133–147], Theorem 1.1, which requires $\unicode[STIX]{x1D702}\in (\frac{32}{17},2.0606\cdots \,)$ in place of $\unicode[STIX]{x1D702}\in (\frac{32}{17},2.1426\cdots \,)$.

On values taken by the largest prime factor of shifted primes (II)

“…As in [9], the letters p, q, r and are always used to denote prime numbers, and d, m, and n always denote positive integers. In what follows, let a ∈ Z * and η ∈ [η 0 , 1 + 4 √ e).…”

“…Moreover for 2 η < 1 + 3 4 √ 2, this estimate holds for any constant c > 1. Very recently, Wu [9] extended Banks-Shparlinski's interval ( 32 17 , 1 + 3…”

“…Proof. The first inequality is Proposition 3.2 of [9]. With the help of Motohashi's convolution argument [7], (2.8) follows from (2.7) and the classical Bombieri-Vinogradov theorem (see (2.6) below).…”

“…For comparison, we have The improvement of Wu [9] comes from the following two simple observations: (i) In many arithmetic applications, the linear sieve is more powerful than the sieve of dimension 2; (ii) With the help of the Chen-Iwaniec switching principle [2,3] and his theorem of Bombieri-Vinogradov type (see [9,Proposition 3.2] or (2.7) below), Wu can sieve the sequence of convolution:…”

Elliott-Halberstam conjecture and values taken by the largest prime factor of shifted primes

“…Moreover for 2 η < 1 + 3 4 √ 2 ≈ 2.0606, this estimate holds for any constant c > 1. Very recently, Wu [12] extended Banks-Shparlinski's interval ( 32 17 , 1 + 3 4 √ 2) to ( 32 17 , η 0 ), where η 0 ≈ 2.142 is the unique solution of the equation η − 1 − 4η log(η − 1) = 0 in (1, ∞). Banks & Shparlinski [1, page 144] also remarked that the asymptotic formula (1.1) holds for η ∈ (1, 32 17 ] if we assume the Elliott-Halberstam conjecture (see EH prime [ε] below).…”