Browse Groups

• ## Re: [PrimeNumbers] Generalised Fermat numbers

(2)
• NextPrevious
• In a message dated 08/04/04 02:37:42 GMT Daylight Time, ... In August 2000 I did a complete search for primes of the form a^(2^n)+b^(2^n) for 2
Message 1 of 2 , Apr 8, 2004
View Source
In a message dated 08/04/04 02:37:42 GMT Daylight Time,
fitzhughrichard@... writes:

> Let Gfp(n) = the pair (a,b) leading to the smallest prime of the form
> a^(2^n)+b^(2^n).
>
> Gfp(1) = (1,2) [1 digit]
> Gfp(2) = (1,2) [2 digits]
> Gfp(3) = (1,2) [3 digits]
> Gfp(4) = (1,2) [5 digits]
> Gfp(5) = (8,9) [31 digits]
> Gfp(6) = (8,11) [68 digits]
> Gfp(7) = (20,27) [184 digits]
> Gfp(8) = (5,14) [294 digits]
> Gfp(9) = (2,13) [571 digits]
> Gfp(10) = (26,47) [1713 digits] (PRP only)
> Gfp(11) = (3,22) [2750 digits] (PRP only)
> Gfp(12) = ? [ > 5700 digits]
>
> How far is Gfp(n) known?
>

In August 2000 I did a complete search for primes of the form a^(2^n)+b^(2^n)
for 2<=n<=13 and a<b<=100, so I can confirm your results and supply the next
2 data points.

Gfp(12) = (2,53) [7063 digits] (PRP only)

Gfp(13) = (43,72) [15216 digits] (PRP only)
This is in Henri Lifchitz's "Top 5000 PRP" site
http://ourworld.compuserve.com/homepages/hlifchitz/
as
<A HREF="http://www.primenumbers.net/prptop/detailprp.php?rank=951">951</A> 72^8192+43^8192 15216 Mike Oakes 08/2000

>Also, is there an efficient primality test for numbers of this form?

None is known.

-Mike Oakes

[Non-text portions of this message have been removed]
Your message has been successfully submitted and would be delivered to recipients shortly.
• Changes have not been saved
Press OK to abandon changes or Cancel to continue editing
• Your browser is not supported
Kindly note that Groups does not support 7.0 or earlier versions of Internet Explorer. We recommend upgrading to the latest Internet Explorer, Google Chrome, or Firefox. If you are using IE 9 or later, make sure you turn off Compatibility View.