From: "Peter L. Montgomery" <Peter-Lawrence.Montgomery@cwi.nl>
Subject: Re: Existence of such primes?
Date: Sun, 30 Jul 2000 14:59:53 GMT
Newsgroups: sci.math
Summary: [missing]

In article <25eg5.197$vH3.26580@news3.cableinet.net> 
"James Wanless" <james@jwanless.freeserve.co.uk> writes:
>Hmmm, not sure it's so easy, actually...  :-(
>However I would venture a generalization:
>2a^n +/- 1    [over all n]
>contains an infinite number of primes for any a>1
>Can anyone confirm/deny this? (seems - at least superficially - very
>interesting - to me anyrate)


      Not necessarily.


      Choose a so that

          a == 2 (mod 3*5*17*257*65537*641)
          a == -2 (mod 6700417)

If n is even, then a^n == 1 (mod 3) and 3 divides 2*a^n + 1.
If n == 1 (mod 4), then a^n == 2 (mod 5) and 5 divides 2*a^n + 1.
If n == 3 (mod 8), then 17 divides 2*a^n + 1.
If n == 7 (mod 16), then 257 divides 2*a^n + 1.
If n == 15 (mod 32), then 65537 divides 2*a^n + 1.
If n == 31 (mod 64) then 641 divides 2*a^n + 1.
If n == 63 (mod 64) then 6700417 divides 2*a^n + 1.

    That is, 2*a^n + 1 is never prime (we can check 2*a^n + 1 > 6700417).
-- 
E = m c^2.  Einstein = Man of the Century.  Why the squaring?

        Peter-Lawrence.Montgomery@cwi.nl    Home: San Rafael, California
        Microsoft Research and CWI
==============================================================================\

From: Foghorn Leghorn <foghorn@netcene.com>
Subject: Re: Existence of such primes?
Date: 28 Jul 2000 20:41:40 -0500
Newsgroups: sci.math

On Fri, 28 Jul 2000 14:26:29 +0800, "Andy" <ckhandy@hongkong.com>
wrote:

>Are there infinitely many prime numbers p of the form p= 2x9^n+1, where n is
>a positive integer?
>For n=1, p=2x9+1=19 (prime)
>For n=2, p=2x81+1=163 (prime)
>For n=3, p=2x729+1=1459 (prime)
>......
>

The function 2*9^n+1 generates primes for the following values of n:
1, 2, 3, 8, 15, 27, 30, 66, 90, 160, 348, 391, 411, 626, 727 -- and
none others below 2500. This didn't take long to check with the
PrimeForm program.

As for whether this function generates infinitely many primes, I would
guess that the question is open. The analagous problems for generating
functions such as n^2+1 and 2^n-1 are still unsovled. See the links

http://www.utm.edu/research/primes/notes/conjectures/index.html

and

http://www.utm.edu/research/primes/mersenne.shtml

Foghorn Leghorn
foghorn@netcene.com