Legendre showed that there is no Rational algebraic function which always gives Primes. In 1752, Goldbach showed that no Polynomial with Integer Coefficients can give a Prime for all integral values. However, there exists a Polynomial in 10 variables with Integer Coefficients such that the set of Primes equals the set of Positive values of this Polynomial obtained as the variables run through all Nonnegative Integers, although it is really a set of Diophantine Equations in disguise (Ribenboim 1991).

Range | Consecutive | Reference | |

[0, 44] | 45 | Fung and Ruby (Mollin and Williams 1990) | |

[0, 42] | 43 | Fung and Ruby (Mollin and Williams 1990) | |

[0, 39] | 40 | Euler | |

[0, 28] | 29 | Legendre | |

[0, 15] | 16 | Legendre | |

[0, 10] | 11 | ||

[0, 10] | 11 |

The above table gives some low-order polynomials which generate only Primes for the first few Nonnegative values
(Mollin and Williams 1990). The best-known of these formulas is that due to Euler (Euler 1772, Ball and Coxeter 1987). Le
Lionnais (1983) has christened numbers such that the Euler-like polynomial

(1) |

Euler also considered quadratics of the form

(2) |

(3) |

**References**

Abel, U. and Siebert, H. ``Sequences with Large Numbers of Prime Values.'' *Am. Math. Monthly* **100**, 167-169, 1993.

Baker, A. ``Linear Forms in the Logarithms of Algebraic Numbers.'' *Mathematika* **13**, 204-216, 1966.

Baker, A. ``Imaginary Quadratic Fields with Class Number Two.'' *Ann. Math.* **94**, 139-152, 1971.

Ball, W. W. R. and Coxeter, H. S. M. *Mathematical Recreations and Essays, 13th ed.* New York: Dover, p. 60, 1987.

Boston, N. and Greenwood, M. L. ``Quadratics Representing Primes.'' *Amer. Math. Monthly* **102**, 595-599, 1995.

Conway, J. H. and Guy, R. K. ``The Nine Magic Discriminants.'' In *The Book of Numbers.* New York: Springer-Verlag,
pp. 224-226, 1996.

Courant, R. and Robbins, H. *What is Mathematics?: An Elementary Approach to Ideas and Methods, 2nd ed.*
Oxford, England: Oxford University Press, p. 26, 1996.

Euler, L. *Nouveaux Mémoires de l'Académie royale des Sciences.* Berlin, p. 36, 1772.

Forman, R. ``Sequences with Many Primes.'' *Amer. Math. Monthly* **99**, 548-557, 1992.

Garrison, B. ``Polynomials with Large Numbers of Prime Values.'' *Amer. Math. Monthly* **97**, 316-317, 1990.

Hendy, M. D. ``Prime Quadratics Associated with Complex Quadratic Fields of Class Number 2.''
*Proc. Amer. Math. Soc.* **43**, 253-260, 1974.

Le Lionnais, F. *Les nombres remarquables.* Paris: Hermann, pp. 88 and 144, 1983.

Mollin, R. A. and Williams, H. C. ``Class Number Problems for Real Quadratic Fields.''
*Number Theory and Cryptology; LMS Lecture Notes Series* **154**, 1990.

Rabinowitz, G. ``Eindeutigkeit der Zerlegung in Primzahlfaktoren in quadratischen Zahlkörpern.''
*Proc. Fifth Internat. Congress Math.* (Cambridge) **1**, 418-421, 1913.

Ribenboim, P. *The Little Book of Big Primes.* New York: Springer-Verlag, 1991.

Sloane, N. J. A. Sequence A014556 in ``The On-Line Version of the Encyclopedia of Integer Sequences.'' http://www.research.att.com/~njas/sequences/eisonline.html.

Stark, H. M. ``A Complete Determination of the Complex Quadratic Fields of Class Number One.'' *Michigan Math. J.* **14**, 1-27, 1967.

Stark, H. M. ``An Explanation of Some Exotic Continued Fractions Found by Brillhart.''
In *Computers in Number Theory, Proc. Science Research Council Atlas Symposium No. 2 held at Oxford, from 18-23 August, 1969*
(Ed. A. O. L. Atkin and B. J. Birch). London: Academic Press, 1971.

Stark, H. M. ``A Transcendence Theorem for Class Number Problems.'' *Ann. Math.* **94**, 153-173, 1971.

© 1996-9

1999-05-26