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4 results

  • ON A CONJECTURE ON SHIFTED PRIMES WITH LARGE PRIME FACTORS, II

  • Part of
  • YUCHEN DING
  • Journal:
    Bulletin of the Australian Mathematical Society , First View
    Published online by Cambridge University Press:
    13 September 2024, pp. 1-8
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  • Let $\mathcal {P}$ be the set of primes and $\pi (x)$ the number of primes not exceeding x. Let $P^+(n)$ be the largest prime factor of n, with the convention $P^+(1)=1$, and $ T_c(x)=\#\{p\le x:p\in \mathcal {P},P^+(p-1)\ge p^c\}. $ Motivated by a conjecture of Chen and Chen [‘On the largest prime factor of shifted primes’, Acta Math. Sin. (Engl. Ser.) 33 (2017), 377–382], we show that for any c with $8/9\le c<1$,

    $$ \begin{align*} \limsup_{x\rightarrow\infty}T_c(x)/\pi(x)\le 8(1/c-1), \end{align*} $$

    which clearly means that

    $$ \begin{align*} \limsup_{x\rightarrow\infty}T_c(x)/\pi(x)\rightarrow 0 \quad \text{as } c\rightarrow 1. \end{align*} $$

  • On a Theorem of Bombieri, Friedlander, and Iwaniec

  • Daniel Fiorilli
  • Journal:
    Canadian Journal of Mathematics / Volume 64 / Issue 5 / 01 October 2012
    Published online by Cambridge University Press:
    20 November 2018, pp. 1019-1035
    Print publication:
    01 October 2012
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  • In this article, we show to what extent one can improve a theorem of Bombieri, Friedlander and Iwaniec by using Hooley's variant of the divisor switching technique. We also give an application of the theorem in question, which is a Bombieri-Vinogradov type theorem for the Tichmarsh divisor problem in arithmetic progressions.