Properties of Prime Numbers

  • Paul Erdős
  • János Surányi
Part of the Undergraduate Texts in Mathematics book series (UTM)


We have seen two proofs in Chapter 2 showing the uniqueness of prime decompositions of integers (Theorem 1.8). The consequences of that theorem there and also in following chapters make it clear why Theorem 1.8 is called the fundamental theorem. The prime numbers mentioned in that theorem are distributed among the integers in a very peculiar way. One can get a feel for this by looking at the sequence of primes less than 150. We list these primes, writing the differences between consecutive primes below them, and writing those differences that are larger than all the previous differences in boldface.


Natural Number Prime Number Prime Divisor Arithmetic Progression Binomial Coefficient 
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  1. 1.
    W.H. Mills: Bulletin Amer. Math. Soc. 53 (1947), p. 604.MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    W. Sierpiński: Comptes Rendus Acad. Sci., Paris 235 (1952), pp. 1078–1079.zbMATHGoogle Scholar
  3. 3.
    P. Erdős and P. Turán: Bulletin Amer. Math. Soc. 54 (1948), pp. 371–378. They were the first to consider questions relating to d n and the first to present results.CrossRefGoogle Scholar
  4. 4.
    P. Erdős: Publicationes Math. Debrecen 1 (1949–1950), pp. 33–37.Google Scholar
  5. 5.
    H. C. Maier: Michigan Math. Journ. 35 (1988), pp. 323–344.MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    R. A. Rankin: Journ. London Math. Soc. 22 (1947), pp. 226–230.MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    P. Erdős: Bull. Amer. Math. Soc. 54 (1948), pp. 885–889.MathSciNetCrossRefGoogle Scholar
  8. 8.
    H. Maier: Advances in Math. 39 (1981) pp. 257–269.MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    R. C. Baker, G. Harman: The difference between consecutive primes, Proc. London Math. Soc. (3) 72, 1996, pp. 261–280.MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    P. Erdős: Quarterly Journ. of Math., Oxford Ser. 6 (1935), pp. 124–128. To date the largest communicated value of c is C · e γ, where γ is the Euler-Mascheroni constant and C ≈ 1.31. H. Maier and C. Pomerance: Transactions of the Amer. Math. Soc, 322 (1990), pp. 201–237. J. Pintz showed that C can be improved to a constant larger than 2. Verbal communication.CrossRefGoogle Scholar
  11. 11.
    R. A. Rankin: Journ. of the London Math. Soc. 13 (1938), pp. 242–247. The result of footnote 8 is somewhat stronger than this.MathSciNetCrossRefGoogle Scholar
  12. 13.
    P. Erdős: Oberdruk uit Math. B. 7 (1938), pp. 1–2. One can find another proof here as well; see Exercise 5 of this reference.Google Scholar
  13. 15.
    For all the details of the above proof, see E. Landau: Vorlesungen über Zahlentheorie (1927), Volume 1, p. 67.Google Scholar
  14. 17.
    P. Erdős: Acta Litt. Univ. Sci., Szeged, Sect. Math. 5 (1932), pp. 194–198.Google Scholar
  15. 18.
    J. Bertrand needed only the existence of a prime between n and 2n — 2 when n > 4. P. L. Chebyshev proved the existence of such a prime in an even smaller interval.Google Scholar
  16. 19.
    P. Erdős: Journ. London Math. Soc. 9 (1934), pp. 282–288.CrossRefGoogle Scholar
  17. 22.
    B. Riemann: Monatsberichte d. Berliner Acad. d. Wiss. (1859), pp. 671–680.Google Scholar
  18. 23.
    J. Hadamard: Bulletin Soc. Math. France 24 (1896), pp. 199–220.MathSciNetzbMATHGoogle Scholar
  19. 24.
    Ch. de la Vallée-Poussin: Annales Soc. Sci. Bruxelles 20 (1896), pp. 183–256 and pp. 281–297.Google Scholar
  20. 25.
    P. Erdős: Proceedings Nat. Acad. Sci. 35 (1949), pp. 374–384. A. Selberg: Annais of Math. 50 (1949), pp. 305–313. For a detailed form of the first paper see Gy. Hoffmann, L. Surányp. Matematikai Lapok 23 (1972), pp. 31–51 (in Hungarian).CrossRefGoogle Scholar
  21. 26.
    For more details, see the second part of Ch. de la Vallée-Poussin’s paper from footnote 8.Google Scholar
  22. 27.
    The proofs of these two Statements in the general case can both be found in E. Landau: Handbuch der Lehre von der Verteilung der Primzahlen, 1909, B. G. Teubner, pp. 436–446.Google Scholar

Copyright information

© Springer Science+Business Media New York 2003

Authors and Affiliations

  • Paul Erdős
  • János Surányi
    • 1
  1. 1.Department of Algebra and Number TheoryEõtvõs Loránd UniversityBudapestHungary

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