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ON VALUES TAKEN BY THE LARGEST PRIME FACTOR OF SHIFTED PRIMES
- Part of
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- Journal:
- Journal of the Australian Mathematical Society / Volume 107 / Issue 1 / August 2019
- Published online by Cambridge University Press:
- 15 August 2018, pp. 133-144
- Print publication:
- August 2019
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- Article
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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 has relative asymptotic density one in
$$\begin{eqnarray}\{p\in \mathbb{P}:p=P(q-a)\text{ for some prime }q\text{ with }p^{\unicode[STIX]{x1D702}}<q\leq c(\unicode[STIX]{x1D702})p^{\unicode[STIX]{x1D702}}\}\end{eqnarray}$$
$\mathbb{P}$. 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 
$\unicode[STIX]{x1D702}\in (\frac{32}{17},2.1426\cdots \,)$.
