## Generalised Fermat numbers

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• I m sure this is covered somewhere, but... 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]
Message 1 of 2 , Apr 7 12:41 PM
I'm sure this is covered somewhere, but...

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?

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

Richard
• 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 2 of 2 , Apr 8 12:23 AM
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

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