The Number of Primes Below a Given Limit

  • Hans Riesel
Part of the Progress in Mathematics book series (PM, volume 126)

Abstract

Consider the positive integers 1,2,3,4, ... Among them there are composite numbers and primes. A composite number is a product of several factors ≠ 1, such as 15 = 3 · 5; or 16 = 2 · 8. A prime p is characterized by the fact that its only possible factorization apart from the order of the factors is p = 1 · p. Every composite number can be written as a product of primes, such as 16 = 2·2·2·2.— Now, what can we say about the integer 1? Is it a prime or a composite? Since 1 has the only possible factorization 1 · 1 we could agree that it is a prime. We might also consider the product 1 · p as a product of two primes; somewhat awkward for a prime number p.—The dilemma is solved if we adopt the convention of classifying the number 1 as neither prime nor composite. We shall call the number 1 a unit. The positive integers may thus be divided into:
  1. 1.

    The unit 1.

     
  2. 2.

    The prime numbers 2, 3, 5, 7, 11, 13, 17, 19, 23,...

     
  3. 3.

    The composite numbers 4, 6, 8, 9, 10, 12, 14, 15, 16,...

     

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Bibliography

  1. 1.
    D. N. Lehmer, Factor Table For the First Ten Millions Containing the Smallest Factor of Every Number Not Divisible By 2, 3, 5, Or 7 Between the Limits 0 and 10,017,000, Hafner, New York, 1956. (Reprint)Google Scholar
  2. 2.
    D. N. Lehmer, List of Prime Numbers From 1 to 10,006,721, Hafner, New York, 1956. (Reprint)Google Scholar
  3. 3.
    C. L. Baker and F. J. Gruenberger, The First Six Million Prime Numbers, The Rand Corporation, Santa Monica. Published by Microcard Foundation, Madison, Wisconsin, 1959.Google Scholar
  4. 4.
    C. L. Baker and F. J. Gruenberger, Primes in the Thousandth Million, The Rand Corporation, Santa Monica, 1958.Google Scholar
  5. 5.
    M. F. Jones, M. Lal and W. J. Blundon, “Statistics on Certain Large Primes,” Math. Comp. 21 (1967) pp. 103–107.MathSciNetMATHCrossRefGoogle Scholar
  6. 6.
    Carter Bays and Richard H. Hudson, “On the Fluctuations of Littlewood for Primes of the Form 4n ± 1,” Math. Comp. 32 (1978) pp. 281–286.MathSciNetMATHGoogle Scholar
  7. 6’.
    A. M. Odlyzko, “Iterated Absolute Values of Differences of Consecutive Primes,” Math. Comp. 61 (1993) pp. 373–380.MathSciNetMATHCrossRefGoogle Scholar
  8. 7.
    A. M. Legendre, Théorie des Nombres, 3 edition, Paris, 1830. Vol. 2, p. 65.Google Scholar
  9. 8.
    D. H. Lehmer, “On the Exact Number of Primes Less Than a Given Limit,” Ill. Journ. Math. 3 (1959) pp. 381–388. Contains many references to earlier work.MathSciNetMATHGoogle Scholar
  10. 9.
    David C. Mapes, “Fast Method for Computing the Number of Primes Less Than a Given Limit,” Math. Comp. 17 (1963) pp. 179–185.MathSciNetMATHCrossRefGoogle Scholar
  11. 10.
    J. C. Lagarias, V. S. Miller and A. M. Odlyzko, “Computing π(x): The Meissel-Lehmer Method,” Math. Comp. 44 (1985) pp. 537–560.MathSciNetMATHGoogle Scholar
  12. 11.
    J. C. Lagarias and A. M. Odlyzko, “Computing π(x): An Analytic Method,” Journ. of Algorithms 8 (1987) pp. 173–191.MathSciNetMATHCrossRefGoogle Scholar
  13. 12.
    Jan Bohman, “On the Number of Primes Less Than a Given Limit,” Nordisk Tidskrift för Informationsbehandling (BIT) 12 (1972) pp. 576–577.MathSciNetMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 1994

Authors and Affiliations

  • Hans Riesel
    • 1
  1. 1.Department of MathematicsThe Royal Institute of TechnologyStockholmSweden

Personalised recommendations