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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 Diophantine Equation Fibonacci Number Algebraic Integer Positive Inte 
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.

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References

  1. 1912.
    FROBENIUS, F.G. Über quadratische Formen, die viele Primzahlen darstellen. Sitzungsber. d. Königl. Akad. d. Wiss. Zu 208 Bibliography Berlin, 1912, 966–980. Reprinted inGesammelte Abhandlungen, Vol. III, 573–587. Springer-Verlag, Berlin, 1968.Google Scholar
  2. 1912.
    RABINOVITCH, G. Eindeutigkeit der Zerlegung in Primzahlfaktoren in quadratischen Zahlkörpern. Proc. Fifth Intern. Congress Math., Cambridge, Vol. 1, 1912, 418–421.Google Scholar
  3. 1938.
    SKOLEM, T. Diophantische Gleichungen. Springer-Verlag, Berlin, 1938.MATHGoogle Scholar
  4. 1947.
    MILLS, W.H. A prime-representing function. Bull. Amer. Math. Soc., 53, p. 604.Google Scholar
  5. 1951.
    WRIGHT, E.M. A prime-representing function. Amer. Math. Monthly, 58, 1951, 616–618.MATHCrossRefGoogle Scholar
  6. 1952.
    HEEGNER, K. Diophantische Analysis und Modulfunktionen. Math. Zeits., 56, 1952, 227–253.MathSciNetMATHCrossRefGoogle Scholar
  7. 1960.
    PUTNAM, H. An unsolvable problem in number theory. J. Symb. Logic, 1960, 220–232.Google Scholar
  8. 1962.
    COHN, H. Advanced Number Theory. Wiley, New York, 1962. Reprinted by Dover, New York, 1980.Google Scholar
  9. 1964.
    WILLANS, C.P. On formulae for the nth prime. Math. Gaz., 48, 1964, 413–415.MathSciNetMATHCrossRefGoogle Scholar
  10. 1966.
    BAKER, A. Linear forms in the logarithms of algebraic numbers. Mathematika, 13, 1966, 204–216.CrossRefGoogle Scholar
  11. 1967.
    STARK, H.M. A complete determination of the complex quadratic fields of class-number one. Michigan Math. J., 14, 1967, 1–27.Google Scholar
  12. 1969.
    DUDLEY, U. History of a formula for primes. Amer. Math. Monthly, 76, 1969, 23–28.MathSciNetMATHCrossRefGoogle Scholar
  13. 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
  14. 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.Google Scholar
  15. 1972.
    VANDEN EYNDEN, C. A proof of Gandhi’s formula for the nth prime. Amer. Math. Monthly, 79, 1972, p. 625.MathSciNetMATHCrossRefGoogle Scholar
  16. 1973.
    DAVIS, M. Hilbert’s tenth problem is unsolvable. Amer. Math. Monthly, 80, 1973, 233–269.MATHCrossRefGoogle Scholar
  17. 1973.
    KARST, E. New quadratic forms with high density of primes. Elem. d. Math., 28, 1973, 116–118.MathSciNetMATHGoogle Scholar
  18. 1974.
    GOLOMB, S.W. A direct interpetation of Gandhi’s formula. Amer. Math. Monthly, 81, 1974, 752–754.MathSciNetMATHCrossRefGoogle Scholar
  19. 1974.
    HENDY, M.D. Prime quadratics associated with complex quadratic fields of class number two. Proc. Amer. Math. Soc., 43, 1974, 253–260.MathSciNetMATHGoogle Scholar
  20. 1975.
    ERNVALL, R. A formula for the least prime greater than a given integer. Elem. d. Math., 30, 1975, 13–14.MathSciNetMATHGoogle Scholar
  21. 1975.
    JONES, J.P. Diophantine representation of the Fibonacci numbers. Fibonacci Quart., 13, 1975, 84–88.MathSciNetMATHGoogle Scholar
  22. 1975.
    MATIJASEVIC, Yu.V. Reduction of an arbitrary diophantine equation to one in 13 unknowns. Acta Arithm., 27, 1975, 521–553.MathSciNetMATHGoogle Scholar
  23. 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
  24. 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.Google Scholar
  25. 1979.
    JONES, J.P. Diophantine representation of Mersenne and Fermat primes, Act. Arith., 35, 1979, 209–221.MATHGoogle Scholar
  26. 1988.
    RIBENBOIM, P. Euler’s famous prime generating polynomial and the class number of imaginary quadratic fields. L’Enseign. Math., 34, 1988, 23–42.MathSciNetMATHGoogle Scholar
  27. 1989.
    GOETGHELUCK, P. On cubic polynomials giving many primes. Elem. d. Math., 44, 1989, 70–73.MathSciNetMATHGoogle Scholar

Copyright information

© Paulo Ribenboim 1991

Authors and Affiliations

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

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