The Michigan Mathematical Journal

Smooth values of shifted primes in arithmetic progressions

William D. Banks, Asma Harcharras, and Igor E. Shparlinski

Full-text: Open access

Article information

Michigan Math. J., Volume 52, Issue 3 (2004), 603-618.

First available in Project Euclid: 16 November 2004

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 11N25: Distribution of integers with specified multiplicative constraints
Secondary: 11N13: Primes in progressions [See also 11B25] 11N36: Applications of sieve methods


Banks, William D.; Harcharras, Asma; Shparlinski, Igor E. Smooth values of shifted primes in arithmetic progressions. Michigan Math. J. 52 (2004), no. 3, 603--618. doi:10.1307/mmj/1100623415.

Export citation


  • W. R. Alford, A. Granville, and C. Pomerance, There are infinitely many Carmichael numbers, Ann. of Math. (2) 139 (1994), 703--722.
  • R. C. Baker and G. Harman, Shifted primes without large prime factors, Acta Arith. 83 (1998), 331--361.
  • N. de Bruijn, On the number of positive integers $\le x$ and free of prime factors $>y,$ Nederl. Akad. Wetensch. Proc. Ser. A 54 (1951), 50--60.
  • C. Dartyge, G. Martin, and G. Tenenbaum, Polynomial values free of large prime factors, Period. Math. Hungar. 43 (2001), 111--119.
  • K. Dickman, On the frequency of numbers containing prime factors of a certain relative magnitude, Ark. Mat. Astr. Fys. 22 (1930), 1--14.
  • P. Erdős, On the normal number of prime factors of $p-1$ and some other related problems concerning Euler's $\phi$-function, Quart. J. Math. Oxford Ser. (2) 6 (1935), 205--213.
  • É. Fouvry and G. Tenenbaum, Entiers sans grand facteur premier en progressions arithmetiques, Proc. London Math. Soc. (3) 63 (1991), 449--494.
  • ------, Répartition statistique des entiers sans grand facteur premier dans les progressions arithmétiques, Proc. London Math. Soc. (3) 72 (1996), 481--514.
  • A. Granville, Integers, without large prime factors, in arithmetic progressions. I, Acta Math. 170 (1993), 255--273.
  • ------, Integers, without large prime factors, in arithmetic progressions. II, Philos. Trans. Roy. Soc. London Ser. A 345 (1993), 349--362.
  • ------, Smooth numbers: Computational number theory and beyond, Proc. MSRI Conf. Algorithmic Number Theory: Lattices, Number Fields, Curves, and Cryptography (Berkeley, 2000) (J. Buhler, P. Stevenhagen, eds.), Cambridge Univ. Press (to appear).
  • H. Halberstam and H.-E. Richert, Sieve methods, London Math. Soc. Monogr., 4, Academic Press, London, 1974.
  • A. Hildebrand, On the number of positive integers $\le x$ and free of prime factors $>y,$ J. Number Theory 22 (1986), 289--307.
  • G. Martin, An asymptotic formula for the number of smooth values of a polynomial, J. Number Theory 93 (2002), 108--182.
  • P. Moree, A note on Artin's conjecture, Simon Stevin 67 (1993), 255--257.
  • C. Pomerance, Popular values of Euler's function, Mathematika 27 (1980), 84--89.
  • C. Pomerance and I. Shparlinski, Smooth orders and cryptographic applications, Algorithmic number theory (Sydney, 2002) (C. Fieker, D. Kohel, eds.), Lecture Notes in Comput. Sci., 2369, pp. 338--348, Springer-Verlag, Berlin, 2002.
  • K. Prachar, Primzahlverteilung, Springer-Verlag, Berlin, 1957.
  • G. Tenenbaum, Introduction to analytic and probabilistic number theory, Cambridge Univ. Press, 1995.
  • S. A. Vanstone and R. J. Zuccherato, Short RSA keys and their generation, J. Cryptology 8 (1995), 101--114.