Journal of The Australian Mathematical Society | 2019
ON VALUES TAKEN BY THE LARGEST PRIME FACTOR OF SHIFTED PRIMES
Abstract
Denote by $\\mathbb{P}$\n the set of all prime numbers and by $P(n)$\n the largest prime factor of positive integer $n\\geq 1$\n with the convention $P(1)=1$\n . In this paper, we prove that, for each $\\unicode[STIX]{x1D702}\\in (\\frac{32}{17},2.1426\\cdots \\,)$\n , there is a constant $c(\\unicode[STIX]{x1D702})>1$\n such that, for every fixed nonzero integer $a\\in \\mathbb{Z}^{\\ast }$\n , the set $$\\begin{eqnarray}\\{p\\in \\mathbb{P}:p=P(q-a)\\text{ for some prime }q\\text{ with }p^{\\unicode[STIX]{x1D702}}<q\\leq c(\\unicode[STIX]{x1D702})p^{\\unicode[STIX]{x1D702}}\\}\\end{eqnarray}$$\n has relative asymptotic density one in $\\mathbb{P}$\n . 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 $\\unicode[STIX]{x1D702}\\in (\\frac{32}{17},2.0606\\cdots \\,)$\n in place of $\\unicode[STIX]{x1D702}\\in (\\frac{32}{17},2.1426\\cdots \\,)$\n .