Conference rusure::math

874.0. "Prime generating polynomials" by CLT::GILBERT () Tue May 17 1988 12:43

>Chris Long writes:
>
>Does anyone have more information concerning consecutive prime values
>of quadratic polynomials with integer coeffiecients?  What is the
>current record?  Is it P(x) = x^2 - 79x + 1601?
>--
 
     The quadratic polynomial: P(x) = x^2 + x + 41  generates 40 successive
primes when x takes on the values 0 to 39, however gives 1681 when x = 40.
The values of x from -40 to -1 also produce primes, but these merely
duplicate those of the previous set.  If we substitute x = y - 40 in P(x),
we get: P(y) = y^2 - 79y + 1601  [Chris mentions].  This generates 80
succesive primes, but only 40 of them are distinct, 2-fold degeneracy.
 
     If a polynomial of the quadratic form,  x^2 + x + b, is to yield
primes for b - 1 succesive values of x, the b must exceed 1,250,000,000
if a b exists.
 
     Other formulas of interest are:  x^3 + x^2 + 17  which yields primes
for x = -14 to +10,  and x^2 - 2999x + 2248541, which yields primes for
the 80 consecutive values of x between 1460 and 1539.  Formulas generating
a disproportionate number of primes for x < 100 are:  2x^2 + 29,
6x^2 + 6x + 31, and 3x^2 + 3x +23.  See, for example, _Recreations_in_the_
Theory_of_Numbers_ by A. H. Beiler.
 
     Wayne K. Schroll         wayne@newton.physics.purdue.edu

Newsgroups: sci.math,sci.crypt
Path: decwrl!labrea!rutgers!iuvax!pur-ee!pur-phy!wayne
Subject: Prime generating polynomials
Posted: 15 May 88 15:40:32 GMT
Organization: Purdue Univ. Physics Dept., W. Lafayette, IN
Xref: decwrl sci.math:3866 sci.crypt:1198

     Presumably the part about x^2 + x + b being prime for
     b - 1 successive values of x [implies b > 1.25E09] means
     for b > 41 [the example from its previous paragraph].
     
     By the way, x^2 + x + b evaluated at x = b - 1 yields the
     nonprime b^2.  The b - 1 successive values of x for b = 41
     are x = 0, ..., b-2.
     
     Dan