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Are There Functions Defining Prime Numbers?

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The Little Book of Big Primes
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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.

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References

  1. 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. RABINOVITCH, G. Eindeutigkeit der Zerlegung in Primzahlfaktoren in quadratischen Zahlkörpern. Proc. Fifth Intern. Congress Math., Cambridge, Vol. 1, 1912, 418–421.

    Google Scholar 

  3. SKOLEM, T. Diophantische Gleichungen. Springer-Verlag, Berlin, 1938.

    MATH  Google Scholar 

  4. MILLS, W.H. A prime-representing function. Bull. Amer. Math. Soc., 53, p. 604.

    Google Scholar 

  5. WRIGHT, E.M. A prime-representing function. Amer. Math. Monthly, 58, 1951, 616–618.

    Article  MATH  Google Scholar 

  6. HEEGNER, K. Diophantische Analysis und Modulfunktionen. Math. Zeits., 56, 1952, 227–253.

    Article  MathSciNet  MATH  Google Scholar 

  7. PUTNAM, H. An unsolvable problem in number theory. J. Symb. Logic, 1960, 220–232.

    Google Scholar 

  8. COHN, H. Advanced Number Theory. Wiley, New York, 1962. Reprinted by Dover, New York, 1980.

    Google Scholar 

  9. WILLANS, C.P. On formulae for the nth prime. Math. Gaz., 48, 1964, 413–415.

    Article  MathSciNet  MATH  Google Scholar 

  10. BAKER, A. Linear forms in the logarithms of algebraic numbers. Mathematika, 13, 1966, 204–216.

    Article  Google Scholar 

  11. 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. DUDLEY, U. History of a formula for primes. Amer. Math. Monthly, 76, 1969, 23–28.

    Article  MathSciNet  MATH  Google Scholar 

  13. 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. 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. VANDEN EYNDEN, C. A proof of Gandhi’s formula for the nth prime. Amer. Math. Monthly, 79, 1972, p. 625.

    Article  MathSciNet  MATH  Google Scholar 

  16. DAVIS, M. Hilbert’s tenth problem is unsolvable. Amer. Math. Monthly, 80, 1973, 233–269.

    Article  MATH  Google Scholar 

  17. KARST, E. New quadratic forms with high density of primes. Elem. d. Math., 28, 1973, 116–118.

    MathSciNet  MATH  Google Scholar 

  18. GOLOMB, S.W. A direct interpetation of Gandhi’s formula. Amer. Math. Monthly, 81, 1974, 752–754.

    Article  MathSciNet  MATH  Google Scholar 

  19. HENDY, M.D. Prime quadratics associated with complex quadratic fields of class number two. Proc. Amer. Math. Soc., 43, 1974, 253–260.

    MathSciNet  MATH  Google Scholar 

  20. ERNVALL, R. A formula for the least prime greater than a given integer. Elem. d. Math., 30, 1975, 13–14.

    MathSciNet  MATH  Google Scholar 

  21. JONES, J.P. Diophantine representation of the Fibonacci numbers. Fibonacci Quart., 13, 1975, 84–88.

    MathSciNet  MATH  Google Scholar 

  22. MATIJASEVIC, Yu.V. Reduction of an arbitrary diophantine equation to one in 13 unknowns. Acta Arithm., 27, 1975, 521–553.

    MathSciNet  MATH  Google Scholar 

  23. JONES, J.P., SATO, D., WADA, H. & WIENS, D. Diophantine representation of the set of prime numbers. Amer. Math. Monthly, 83, 1976, 449–464.

    Article  MathSciNet  MATH  Google Scholar 

  24. 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. JONES, J.P. Diophantine representation of Mersenne and Fermat primes, Act. Arith., 35, 1979, 209–221.

    MATH  Google Scholar 

  26. RIBENBOIM, P. Euler’s famous prime generating polynomial and the class number of imaginary quadratic fields. L’Enseign. Math., 34, 1988, 23–42.

    MathSciNet  MATH  Google Scholar 

  27. GOETGHELUCK, P. On cubic polynomials giving many primes. Elem. d. Math., 44, 1989, 70–73.

    MathSciNet  MATH  Google Scholar 

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© 1991 Paulo Ribenboim

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Ribenboim, P. (1991). Are There Functions Defining Prime Numbers?. In: The Little Book of Big Primes. Springer, New York, NY. https://doi.org/10.1007/978-1-4757-4330-2_4

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  • DOI: https://doi.org/10.1007/978-1-4757-4330-2_4

  • Publisher Name: Springer, New York, NY

  • Print ISBN: 978-0-387-97508-5

  • Online ISBN: 978-1-4757-4330-2

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