Advertisement

Are There Functions Defining Prime Numbers?

  • Paulo Ribenboim

Abstract

To determine prime numbers, it is natural to ask for functions f(n) defined for all natural numbers n ≥ 1, which are computable in practice and produce some or all prime numbers.

Keywords

Prime Number Class Number Diophantine Equation Algebraic Integer Quadratic Field 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1839.
    DIRICHLET, G.L. 1840 Recherches sur diverses applications de l’analyse infinitésimale la théorie des nombres. Journal f. d. reine u. angew. Math., 19, 1839, 342–369 and 21, 1840, 1–12 and 134–135. Reprinted in Werke (edited by L. Kronecker), Vol. I, 411–496. G. Reimer, Berlin, 1889. Reprinted by Chelsea, Bronx, N.Y., 1969.Google Scholar
  2. 1852.
    KUMMER, E.E. Uber die Erganzungssätze zu den allgemeinem Reciprocitätsgestzen. Journal f. d. reine u. angew. Math., 44, 1852, 93–146. Reprinted in Collected Papers (edited by A. Weil), Vol. I, 485–538. Springer-Verlag, New York, 1975.MATHCrossRefGoogle Scholar
  3. 1912.
    FROBENIUS, F.G. Uber quadratische Formen, die viele Primzahlen darstellen. Sitzungsber. d. Königl. Akad. d. Wiss. zu Berlin, 1912, 966–980. Reprinted in Gesammelte Abhandlungen, vol. III, 573–587. Springer-Verlag, Berlin, 1968.Google Scholar
  4. 1912.
    RABINOVITCH, G. Eindeutigkeit der Zerlegung in Primzahlfaktoren in quadratischen Zahlkörpern. Proc. Fifth Intern. Congress Math., Cambridge, Vol. 1, 1912, 418–421.Google Scholar
  5. 1918.
    LANDAU, E. iber die Klassenzahl imaginär quadratischer Zahlkörper. Göttinger Nachr., 1918, 285–295.Google Scholar
  6. 1933.
    DEURING, M. Imaginär-quadratische Zahlkörper mit Klassenzahl Eins. Invent. Math., 5, 1968, 169–179.MathSciNetMATHCrossRefGoogle Scholar
  7. 1933.
    LEHMER, D.H. On imaginary quadratic fields whose class number is unity. Bull. Amer. Math. Soc., 39, 1933, p. 360.Google Scholar
  8. 1934.
    HEILBRONN, H. On the class number in imaginary quadratic fields. Quart. J. Pure & Appl. Math., Oxford, Ser. 2, 5, 1934, 150–160.CrossRefGoogle Scholar
  9. 1934.
    HEILBRONN, H. & LINFOOT, E.H. On the imaginary quadratic corpora of class-number one. Quart. J. Pure & Appl. Math., Oxford, Ser. 2, 5, 1934, 293–301.CrossRefGoogle Scholar
  10. 1934.
    MORDELL, L.J. On the Riemann hypothesis and imaginary quadratic fields with a given class number. J. London Math. Soc., 9, 1934, 289–298.MathSciNetCrossRefGoogle Scholar
  11. 1935.
    SIEGEL, C.L. Uber die Classenzahl quadratischer Zahlkörper. Acta Arithm., 1, 1935, 83–86. Reprinted in Gesammelte Abhandlungen (edited by K. Chandrasekharan & H. Maaß), Vol. I, 406–409. Springer-Verlag, Berlin, 1966.MATHGoogle Scholar
  12. 1936.
    LEHMER, D.H. On the function x2 + x + A. Sphinx, 6, 1936, 212–214.Google Scholar
  13. 1938.
    SKOLEM, T. Diophantische Gleichungen. Springer-Verlag, Berlin, 1938.MATHGoogle Scholar
  14. 1943.
    REINER, I. Functions not formulas for primes. Amer. Math. Monthly, 50, 1943, 619–621.MathSciNetCrossRefGoogle Scholar
  15. 1946.
    BUCK, R.C. Prime-representing functions. Amer. Math. Monthly, 53, 1946, p. 265.MathSciNetMATHCrossRefGoogle Scholar
  16. 1947.
    MILLS, W.H. A prime-representing function. Bull. Amer. Math. Soc., 53, p. 604.Google Scholar
  17. 1951.
    WRIGHT, E.M. A prime-representing function. Amer. Math. Monthly, 58, 1951, 616–618.MathSciNetMATHCrossRefGoogle Scholar
  18. 1952.
    HEEGNER, K. Diophantische Analysis und Modulfunktionen. Math. Zeits., 56, 1952, 227–253.MathSciNetMATHCrossRefGoogle Scholar
  19. 1952.
    SIERPINSKI, W. Sur une formule donnant tous les nombres premiers. C.R. Acad. Sci. Paris, 235, 1952, 1078–1079.MathSciNetMATHGoogle Scholar
  20. 1960.
    PUTNAM, H. An unsolvable problem in number theory. J. Symb. Logic, 1960, 220–232.Google Scholar
  21. 1962.
    COHN, H. Advanced Number Theory. Wiley, New York, 1962. Reprinted by Dover, New York, 1980.MATHGoogle Scholar
  22. 1964.
    BOREVICH, Z.I. & Shafarevich, I.R. Number Theory. Moscow, 1964. English translation by N. Greenleaf, published by Academic Press, New York, 1966.MATHGoogle Scholar
  23. 1964.
    WILLANS, C.P. On formulae for the nth prime. Math. Gaz., 48, 1964, 413–415.MathSciNetMATHCrossRefGoogle Scholar
  24. 1966.
    BAKER, A. Linear forms in the logarithms of algebraic numbers. Mathematika, 13, 1966, 204–216.CrossRefGoogle Scholar
  25. 1967.
    GOODSTEIN, R.L. & WORMELL, C.P. Formulae for primes. Mat. Gaz., 51, 1967, 35–38.CrossRefGoogle Scholar
  26. 1967.
    STARK, H.M. A complete determination of the complex quadratic fields of class-number one. Michigan Math. J., 14, 1967, 1–27.MathSciNetMATHCrossRefGoogle Scholar
  27. 1968.
    DEURING, M. Imaginäre quadratische Zahlkörper mit der Klassenzahl Eins. Invent. Math., 5, 1968, 169–179.MathSciNetMATHCrossRefGoogle Scholar
  28. 1969.
    BAKER, A. A remark on the class number of quadratic fields. Bull. London Math. Soc., 1, 1969, 98–102.MathSciNetMATHCrossRefGoogle Scholar
  29. 1969.
    DUDLEY, U. History of a formula for primes. Amer. Math. Monthly, 76, 1969, 23–28.MathSciNetMATHCrossRefGoogle Scholar
  30. 1969.
    STARK, H.M. A historical note on complex quadratic fields with class-number one. Proc. Amer. Math. Soc., 21, 1969, 254–255.MathSciNetMATHGoogle Scholar
  31. 1971.
    BAKER, A. Imaginary quadratic fields with class number 2. Annals of Math., 94, 1971, 139–152.MATHCrossRefGoogle Scholar
  32. 1971.
    BAKER, A. On the class number of imaginary quadratic fields. Bull. Amer. Math. Soc., 77, 1971, 678–684.MathSciNetMATHCrossRefGoogle Scholar
  33. 1971.
    GANDHI, J.M. Formulae for the nth prime. Proc. Washington State Univ. Conf. on Number Theory, 96–106. Wash. St. Univ., Pullman, Wash., 1971.Google Scholar
  34. 1971.
    MATIJASEVIC, Yu.V. Diophantine representation of the set of prime numbers (in Russian). Dokl. Akad. Nauk SSSR, 196, 1971, 770–773. English translation by R. N. GOSS, in Soviet Math. Dokl. 11, 1970, 354–358.MathSciNetGoogle Scholar
  35. 1972.
    SIEGEL, C.L. Zur Theorie der quadratischen Formen. Nachr. Akad. Wiss. Göttingen, Math. Phys. K1., 1972, No. 3, 21–46. Reprinted in Gesammelte Abhandlungen (edited by K. Chandrasekharan & H. Maaß), Vol. IV, 224–249. Springer-Verlag, New York, 1979.Google Scholar
  36. 1972.
    VANDEN EYNDEN, A proof of Gandhi’s formula C. for the nth prime. Amer. Math. Monthly, 79, 1972, p. 625.MathSciNetMATHCrossRefGoogle Scholar
  37. 1973.
    DAVIS, M. Hilbert’s tenth problem is unsolvable. Amer. Math. Monthly, 80, 1973, 233–269.MATHCrossRefGoogle Scholar
  38. 1973.
    KARST, E. New quadratic forms with high density of primes. Elem. d. Math., 28, 1973, 116–118.MathSciNetMATHGoogle Scholar
  39. 1974.
    GOLOMB, S.W. A direct interpretation of Gandhi’s formula. Amer. Math. Monthly, 81, 1974, 752–754.MathSciNetMATHCrossRefGoogle Scholar
  40. 1974.
    HENDY, M.D. Prime quadratics associated with complex quadratic fields of class number two. Proc. Amer. Math. Soc., 43, 1974, 253–260.MathSciNetMATHCrossRefGoogle Scholar
  41. 1974.
    MONTGOMERY, H.L. & Weinberger, P.J. Notes on small class numbers Acta Arithm., 24, 1974, 529–542.MathSciNetMATHGoogle Scholar
  42. 1974.
    SZEKERES, G. On the number of divisors of x2 + x + A. J. Nb. Th., 6, 1974, 434–442.MathSciNetMATHGoogle Scholar
  43. 1975.
    BAKER, A. Transcendental Number Theory. Cambridge Univ. Press, Cambridge, 1975.MATHCrossRefGoogle Scholar
  44. 1975.
    ERNVALL, R. A formula for the least prime greater than a given integer. Elem. d. Math., 30, 1975, 13–14.MathSciNetMATHGoogle Scholar
  45. 1975.
    JONES, J.P. Diophantine representation of the Fibonacci numbers. Fibonacci Quart., 13, 1975, 84–88.MathSciNetMATHGoogle Scholar
  46. 1975.
    MATIJASEVIC, Yu.V. Reduction of an arbitrary diophantine equation to one in 13 unknowns. Acta Arithm., 27, 1975, 521–553.MathSciNetMATHGoogle Scholar
  47. 1975.
    STARK, H.M. On complex quadratic fields with class-number two. Math. Comp., 29, 1975, 289–302.MathSciNetMATHGoogle Scholar
  48. 1976.
    GOLDFELD, D. The class number of quadratic fields and conjectures of Birch and Swinnerton-Dyer. Ann. Scuola Norm. Sup. Pisa, Ser. 4, 3, 1976, 623–663.Google Scholar
  49. 1976.
    JONES, J.P., SATO, D., WADA, H. & WIENS, D. Diophantine representation of the set of prime numbers. Amer. Math. Monthly, 83, 1976, 449–464.MathSciNetMATHCrossRefGoogle Scholar
  50. 1977.
    MATIJASEVIC, Yu.V. Primes are nonnegative values of a polynomial in 10 variables. Zapiski Sem. Leningrad Mat. Inst. Steklov, 68, 1977, 62–82. English translation by L. Guy & J.P. Jones, J. Soviet Math., 15, 1981, 33–44.MathSciNetGoogle Scholar
  51. 1977.
    MATIJASEVIC, Yu.V. Some purely mathematical results inspired by mathematical logic, in Logic, Foundations of Mathematics and Computability Theory (Proc. 5th Intern. Congress Logic, Methodology and Philosophy of Science, Univ. Western Ont., London, Ont., 1975), 121–127. Reidel, Dordrecht, 1977.CrossRefGoogle Scholar
  52. 1979.
    JONES, J.P. Diophantine representation of Mersenne and Fermat primes. Acta Arithm., 35, 1979, 209–221.MATHGoogle Scholar
  53. 1981.
    AYOUB, R.G. & CHOWLA, S. On Euler’s polynomial. J. Nb. Th., 13, 1981, 443–445.MathSciNetMATHGoogle Scholar
  54. 1981.
    HATCHER, W.S. & HODGSON, B.R. Complexity bounds on proof. J. Symb. Logic, 46, 1981, 255–258.MathSciNetMATHCrossRefGoogle Scholar
  55. 1982.
    JONES, J.P. Universal diophantine equation. J. Symb. Logic, 47, 1982, 549–571.MATHCrossRefGoogle Scholar
  56. 1983.
    GROSS, B. & POINTS de Heegner et dérivées Zagier, D. de fonctions L. C.R. Acad. Sci. Paris, 297, 1983, 85–87.MATHGoogle Scholar
  57. 1984.
    OESTERLE, J. Nombres de classes des corps quadratiques imaginaires. Séminaire Bourbaki, exposé No. 631, 1983/4, Paris. Astérisque, 121/2, 1985, 213–224.MathSciNetGoogle Scholar
  58. 1985.
    GOLDFELD, D. Gauss’ class number problem for imaginary quadratic fields. Bull. Amer. Math. Soc., 13, 1985, 23–37.MathSciNetMATHCrossRefGoogle Scholar
  59. 1987.
    RIBENBOIM, P. Euler’s famous prime generating polynomial and the class number of imaginary quadratic fields. L’Enseign. Math. (to appear). Same in Spanish translation by V. Albis Gonzalez, in Rev. Colombiana de Matematicas (to appear).Google Scholar

Copyright information

© Springer-Verlag New York Inc. 1988

Authors and Affiliations

  • Paulo Ribenboim
    • 1
  1. 1.Department of Mathematics and StatisticsQueen’s UniversityKingstonCanada

Personalised recommendations