On the greatest prime factor of with effective constants

Author:
G. Harman

Journal:
Math. Comp. **74** (2005), 2035-2041

MSC (2000):
Primary 11N13

DOI:
https://doi.org/10.1090/S0025-5718-05-01749-7

Published electronically:
February 16, 2005

MathSciNet review:
2164111

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Let denote a prime. In this article we provide the first published lower bounds for the greatest prime factor of exceeding in which the constants are effectively computable. As a result we prove that it is possible to calculate a value such that for every there is a with the greatest prime factor of exceeding . The novelty of our approach is the avoidance of any appeal to Siegel's Theorem on primes in arithmetic progression.

**1.**M. Agrawal, N. Kayal and N. Saxena,*PRIMES is in P*, http://www.cse.iitk.ac.in/ primality.pdf.**2.**R. C. Baker, G. Harman,*The Brun-Titchmarsh theorem on average*, Analytic Number Theory (Proceedings in honor of Heini Halberstam), Birkhauser, Boston, 1996, 39-103. MR**1399332 (97h:11096)****3.**R. C. Baker, G. Harman,*Shifted primes without large prime factors*, Acta Arith.**83**(1998), 331-361. MR**1610553 (99b:11104)****4.**D. Bernstein,*Proving primality after Agrawal-Kayal-Saxena*, http://cr.yp.to/papers/ html#aks.**5.**E. Bombieri, J. B. Friedlander and H. Iwaniec,*Primes in arithmetic progressions to large moduli III*, J. American Math. Soc.**2**(1989), 215-224. MR**0976723 (89m:11087)****6.**H. Davenport,*Multiplicative Number Theory*(second edition revised by H. L. Montgomery), Springer-Verlag, New York, 1980. MR**0606931 (82m:10001)****7.**J.-M. Deshouillers and H. Iwaniec,*On the Brun-Titchmarsh theorem on average*in Topics in classic number theory (ed. G. Halász), vol. 1 (Budapest, 1981), 319-333. MR**0781145 (86e:11085)****8.**K. Ford,*Vinogradov's Integral and bounds for the Riemann zeta-function*, Proc. London Math. Soc. (3)**85**(2002), 565-633. MR**1936814 (2003j:11089)****9.**M. Goldfeld,*On the number of primes for which has a large prime factor*, Mathematika**16**(1969), 23-27. MR**0244176 (39:5493)****10.**H. L. Montgomery and R. C. Vaughan,*The large sieve*, Mathematika**20**(1973), 119-134. MR**0374060 (51:10260)****11.**Y. Motohashi,*A note on the least prime in an arithmetic progression with a prime difference*, Acta Arith.**17**(1970), 283-285. MR**0268131 (42:3030)****12.**N. M. Timofeev,*The Vinogradov-Bombieri theorem*, (English) Math. Notes**38**(1985), 947-951. MR**0823418 (87f:11073)****13.**R. C. Vaughan,*An elementary method in prime number theory*, Acta Arith.**37**(1980), 111-115. MR**0598869 (82c:10055)**

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Additional Information

**G. Harman**

Affiliation:
Department of Mathematics, Royal Holloway University of London, Egham, Surrey TW20 0EX, United Kingdom

Email:
G.Harman@rhul.ac.uk

DOI:
https://doi.org/10.1090/S0025-5718-05-01749-7

Received by editor(s):
March 19, 2004

Received by editor(s) in revised form:
August 16, 2004

Published electronically:
February 16, 2005

Article copyright:
© Copyright 2005
American Mathematical Society