Generalized Fermat number

There are different kinds of generalized Fermat numbers.

John Cosgrave

John Cosgrave has studied the following numbers:

Numbers of the form: [math]\displaystyle{ F_{n,r} = \sum_{i=0}^{p-1} \ 2^{i p^{n}} \ = \ 2^{(p-1)p^n}+2^{(p-2)p^n}+...+2^{2p^n}+2^{p^n}+1 \ = \ (2^{p^{n+1}}-1)/(2^{p^n}-1) }[/math] where p is the prime of apparition rank r (r(2)=1, r(3)=2, r(5)=3, ...) and n is greater or equal to 0.

[math]\displaystyle{ F_{0,r} }[/math] generates the Mersenne numbers.

[math]\displaystyle{ F_{n,1} }[/math] generates the Fermat numbers.

[math]\displaystyle{ F_{n,2} }[/math] generates the Saouter numbers.

Cosgrave has proven the following properties:

1. If number [math]\displaystyle{ \sum_{i=0}^{p-1}\ (2^i)^{m} \ }[/math] is prime, then [math]\displaystyle{ m=p^n }[/math].
2. [math]\displaystyle{ F_{n,r} }[/math] numbers are pairwise relatively prime within a rank and across ranks: [math]\displaystyle{ gcd(F_{n,i},F_{m,j}) =1 }[/math] for all n, m, i and j.
3. They satisfy a product property like Fermat numbers have. And every [math]\displaystyle{ F_{n,r} }[/math] passes Fermat's test to base 2.

Saouter has proven that [math]\displaystyle{ F_{n,2} }[/math] numbers can be proven prime by using Pépin's test with k=5.

Dubner

In 1985, Dubner for the first time built a list of large primes of the form: b^2m+1, b ≥ 2 and m ≥ 1.

See also: H.Dubner, W.Keller: "Factors of generalized Fermat numbers" (1995)

Björn & Riesel

In 1998, Björn & Riesel for the first time built a list of large primes of the form: a^2m+b^2m, b > a ≥ 2 and m ≥ 1.

Notes for this Wiki

Divisibilities of Generalized Fermat numbers for any Proth primes k•2ⁿ+1 are listed as GF Divisor on their own page. They are listed as F(n), GF(n,a) or xGF(n,a,b) as used by the output of PFGW or at the The Prime Pages.

To test any Proth prime for divisibilities of Generalized Fermat numbers the program PFGW can be used. For example to test Proth 3•2⁴¹+1 call

pfgw -gxo -a2 -q"3*2^41+1"

which results in

3*2^41+1 is a Factor of F38!!!! (0.000000 seconds)
3*2^41+1 is a Factor of GF(38,3)!!!! (0.000000 seconds)
3*2^41+1 is a Factor of xGF(37,3,2)!!!! (0.000000 seconds)
3*2^41+1 is a Factor of xGF(38,4,3)!!!! (0.000000 seconds)
3*2^41+1 is a Factor of GF(35,6)!!!! (0.000000 seconds)
3*2^41+1 is a Factor of xGF(36,8,3)!!!! (0.000000 seconds)
3*2^41+1 is a Factor of xGF(38,9,2)!!!! (0.000000 seconds)
3*2^41+1 is a Factor of xGF(38,9,8)!!!! (0.000000 seconds)
3*2^41+1 is a Factor of xGF(39,11,7)!!!! (0.000000 seconds)
3*2^41+1 is a Factor of GF(38,12)!!!! (0.000000 seconds)
GFN testing completed

listed on the GF Divisor page.

Only proper factors of Generalized Fermat numbers are listed.

Special conditions for Proth primes

A Proth prime written in the normalized form k•bⁿ+1 is also a Generalized Fermat number under special conditions:

If the k-value is a square and the n-value even the Proth prime k•bⁿ+1 is also a Generalized Fermat number of the form

[math]\displaystyle{ \Large (\sqrt{k}\cdot b^{n\over 2})^{2^1} + 1 }[/math].

Example: The Proth prime 289•2¹⁸⁵⁰²+1 is also a Generalized Fermat number of the form (17*2⁹²⁵¹)²¹+1. See 289*2¹⁸⁵⁰²+1 which is also a Cullen prime.

External links

• Generalized Fermat numbers
• Factorization of numbers of the form F_n,2: it includes a program to factor generalized Fermat numbers.
• Factors of generalized Fermat numbers found after Björn & Riesel
• Factors of generalized Fermat numbers found after Björn & Riesel (original)
• MathWorld article
• Generalized Fermat Prime Search
• List of generalized Fermat primes in bases up to 1000
• List of generalized Fermat primes in bases up to 1030

References

Generalized Fermat numbers

Miscellaneous

Generalized Fermat number

Fermat numbers

22n+1 : Fermat numbers Fermat Divisors

Gen. Fermat primes

a2n+1 : n=15 (32768) n=16 (65536) n=17 (131072) n=18 (262144) n=19 (524288) n=20 (1048576)

Gen. Fermat primes

a2n+b2n as (a,b) : (2,1) (3,1) (3,2) (4,3) (5,1) (5,2) (5,3) (5,4) (6,1) (6,5) (7,1) (7,2) (7,3) (7,4) (7,5) (7,6) (8,1) (8,3) (8,5) (8,7) (9,2) (9,5) (9,7) (9,8) (10,1) (10,3) (10,7) (10,9) (11,1) (11,2) (11,3) (11,4) (11,5) (11,6) (11,7) (11,8) (11,9) (11,10) (12,1) (12,5) (12,7) (12,11)

Gen. Fermat primes categories

(2,1) (3,1) (3,2) (4,3) (5,1) (5,2) (5,3) (5,4) (6,1) (6,5) (7,1) (7,2) (7,3) (7,4) (7,5) (7,6) (8,1) (8,3) (8,5) (8,7) (9,2) (9,5) (9,7) (9,8) (10,1) (10,3) (10,7) (10,9) (11,1) (11,2) (11,3) (11,4) (11,5) (11,6) (11,7) (11,8) (11,9) (11,10) (12,1) (12,5) (12,7) (12,11)

GF Divisors

(2,1) (3,1) (3,2) (4,3) (5,1) (5,2) (5,3) (5,4) (6,1) (6,5) (7,1) (7,2) (7,3) (7,4) (7,5) (7,6) (8,1) (8,3) (8,5) (8,7) (9,2) (9,5) (9,7) (9,8) (10,1) (10,3) (10,7) (10,9) (11,1) (11,2) (11,3) (11,4) (11,5) (11,6) (11,7) (11,8) (11,9) (11,10) (12,1) (12,5) (12,7) (12,11)