Baker 1998 shifted primes without large prime factors

../

---

Baker, R. C. and Harman, G., Shifted primes without large prime factors. Acta Arith. 83 (1998), 331--361. The file's text layer carries no copyright or license line; the publisher's record offers the PDF under the link "Pobierz zgodnie z CC-BY" ("Free download under CC-BY license" on the English site) and names no Creative Commons version or URL (https://www.impan.pl/get/doi/10.4064/aa-83-4-331-361, read 2026-10-02), so the term is the Creative Commons Attribution license with its version unstated; the site footer "Copyright © 2026 by IMPAN. All rights reserved." speaks for the site, not the article.

For fixed nonzero a, let Pi(x,y) count primes p <= x with largest prime factor of p-a at most y. Theorem 1 proves Pi(x,y) > x/(log x)^{C_1} for y >= x^beta, beta = 0.2961 and x >= x_0 (x_0 may depend on a, C_1 absolute), improving Friedlander's exponent 1/(2 sqrt e) + eps = 0.3032...; the method shows beta can be lowered slightly further. Two corollaries are drawn: the Erdos-Pomerance result that the integers m with more than m^{1-beta} solutions to Euler's phi(n)=m form an infinite sequence with log m_{i+1}/log m_i tending to 1, and the Alford-Granville-Pomerance bound that the number of Carmichael numbers up to x is at least x^{(5-5beta)/12} for large x. Theorem 2 additionally gives P^+(p-a) > p^{0.677} for infinitely many primes p. The proof counts solutions of p-a = lmn with m,n of size about x^{1-theta} and l a product of many small factors, using Harman's sieve together with the Bombieri-Friedlander-Iwaniec equidistribution results to obtain upper and lower bounds c x L^{-1} sum 1/phi(q) and c' x L^{-1} sum 1/phi(q) for Pi(x,S), with constants c, c' not much greater than one. This supplies the smooth-shifted-prime input behind Erdos problem 821.

Source: http://matwbn.icm.edu.pl/ksiazki/aa/aa83/.

Bears on. #821

Results to transcribe.

• Theorem 1: For y >= x^{0.2961} and x >= x_0 (x_0 may depend on a), the count of primes a<p<=x with P^+(p-a) <= y exceeds x/(log x)^{C_1} for an absolute constant C_1.
• Theorem 2: For infinitely many primes p, P^+(p-a) > p^{0.677}.
• Corollary 1 (Erdos-Pomerance): The integers m with more than m^{1-beta} solutions to phi(n)=m form an infinite sequence with log m_{i+1}/log m_i -> 1.
• Corollary 2 (Alford-Granville-Pomerance): The number of Carmichael numbers up to x is >= x^{(5-5beta)/12} for large x, where beta = 0.2961.