Shifted primes without large prime factors. (English) Zbl 0994.11033

Let \(\pi(x,y)=\sum_{\substack{ a<p\leq x\\ P(p-a)\leq y}} 1\), where \(a\) is fixed, \(p\) denotes a prime, and \(P(m)\) is the largest prime factor of \(m\). The authors establish the lower bound \[ \pi (x,y) > {x \over (\log x)^c} , \] valid for an absolute constant \(c>0\), \(x\geq x_0(a)\), and \(y\geq x^\beta\) with \(\beta = 0.2961\). This improves on the result of J. B. Friedlander [Number theory and applications, Proc. NATO ASI, Banff/Can. 1988, NATO ASI Ser., Ser. C 265, 393-401 (1989; Zbl 0686.10030)], who had any \(\beta>1/2\sqrt 2 e=0.3032\). Earlier results were given by C. Pomerance [Mathematika 27, 84-89 (1980; Zbl 0437.10001)], A. Balog [Sémin. Théor. Nombres, Univ. Bordeaux I 1983/1984, Exp. No. 31, 5 p. (1984; Zbl 0588.10052)], and E. Fouvry and F. Grupp [J. Reine Angew. Math. 370, 101-126 (1986; Zbl 0588.10051)].
There are two immediate corresponding results. Corollary 1 (Erdős-Pomerance). Let \(m_1<m_2<\cdots\) denote the positive integers \(m\) such that the equation \(\varphi(n)=m\) has more than \(m^{1-\beta}\) solutions \(n\). Then the sequence \((m_i)\) is infinite and \(\log m_{i+1} \sim \log m_i\). Corollary 2 (Alford-Granville-Pomerance). The number of Carmichael numbers \(\leq x\) exceeds \(x^{5(1-\beta)/12}\) for large \(x\).
For the problem, Balog had introduced the idea of counting the solutions of the equation \(p-a = mn\), with \(x< p \leq 2x\) and \(x^{1/2 + \delta} < m \leq x^{1/2 + 2\delta}\), and trying to show that the contribution from integers with \(P(mn)\) large cannot deliver all the solutions. Using the Balog-Friedlander construction, the authors add flexibility to the strategy by considering instead the equation \(p-a = \ell mn\), \(x < p \leq 2x\), where \(m\), \(n\) are about \(x^{1/2 - \theta}\), with \(\theta=0.516\), and \(\ell\) is the product of many factors about \(x^\epsilon\) in size. Existing results are then used to bound the relevant sums so derived, but much more effort is required in order to obtain an actual improvement on the earlier arguments.
The authors also sharpen their earlier bound [Berndt, Bruce C. (ed.) et al., Analytic number theory. Vol. 1. Prog. Math. 138, 39-103 (1996; Zbl 0853.11078)] for the corresponding problem with large prime factors. Thus, they prove that \(P(p -a) > p^{0.667}\) for infinitely many primes \(p\), which is the latest improvement in a long line of previous results.


11N25 Distribution of integers with specified multiplicative constraints
11N13 Primes in congruence classes
11N36 Applications of sieve methods
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Greatest prime divisor of prime(n) - 1.