[Search for users]
[Overall Top Noters]
[List of all Conferences]
[Download this site]
| Title: | Mathematics at DEC |
|
| Moderator: | RUSURE::EDP |
|
| Created: | Mon Feb 03 1986 |
| Last Modified: | Fri Jun 06 1997 |
| Last Successful Update: | Fri Jun 06 1997 |
| Number of topics: | 2083 |
| Total number of notes: | 14613 |
874.0. "Prime generating polynomials" by CLT::GILBERT () Tue May 17 1988 09:43
>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 [email protected]
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
| T.R | Title | User | Personal
Name | Date | Lines |
|---|
| 874.1 | by the way | ZFC::DERAMO | I am, therefore I'll think. | Wed May 18 1988 00:05 | 9 |
| 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
|