Generalized Fermat Number
Author
Eric W. Weisstein
Title
Generalized Fermat Number
Description
There are two different definitions of generalized Fermat numbers, one of which is more general than the other. Ribenboim (1996, pp. 89 and 359-360) defines a generalized Fermat number as a number of the form a^(2^n)+1 with a>2, while Riesel (1994) further generalizes, defining it to be a number of the form a^(2^n)+b^(2^n). Both definitions generalize the usual Fermat numbers F_n=2^(2^n)+1. The following table gives the first few generalized Fermat numbers for various bases a. a Sloane...
Category
Educational Materials
Keywords
URL
http://www.notebookarchive.org/2019-07-0z4eh25/
DOI
https://notebookarchive.org/2019-07-0z4eh25
Date Added
2019-07-02
Date Last Modified
2019-07-02
File Size
11.26 kilobytes
Supplements
Rights
Redistribution rights reserved




Generalized Fermat Number
Generalized Fermat Number
Author
Author
Eric W. Weisstein
November 21, 2002
November 21, 2002
This notebook downloaded from http://mathworld.wolfram.com/notebooks/IntegerSequences/GeneralizedFermatNumber.nb.
For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/GeneralizedFermatNumber.html.
©2005 Wolfram Research, Inc. except for portions noted otherwise
Enumeration
Enumeration
Table[a^2^n+1,{a,6},{n,0,6}]//ColumnForm
{2,2,2,2,2,2,2} |
{3,5,17,257,65537,4294967297,18446744073709551617} |
{4,10,82,6562,43046722,1853020188851842,3433683820292512484657849089282} |
{5,17,257,65537,4294967297,18446744073709551617,340282366920938463463374607431768211457} |
{6,26,626,390626,152587890626,23283064365386962890626,542101086242752217003726400434970855712890626} |
{7,37,1297,1679617,2821109907457,7958661109946400884391937,63340286662973277706162286946811886609896461828097} |
Primes
Primes
Select[Table[2^2^n+1,{n,12}],PrimeQ]//Timing
{23.03Second,{5,17,257,65537}}
Select[Table[3^2^n+1,{n,20}],PrimeQ]//Timing
{1.52Second,{}}
Select[Table[4^(2^n)+1,{n,12}],PrimeQ]//Timing
{135.18Second,{17,257,65537}}
Select[Table[5^2^n+1,{n,20}],PrimeQ]//Timing
{2.52Second,{}}
Select[Table[6^(2^n)+1,{n,12}],PrimeQ]//Timing
{167.08Second,{37,1297}}
Select[Table[6^(2^n)+1,{n,13,13}],PrimeQ]//Timing
$Aborted


Cite this as: Eric W. Weisstein, "Generalized Fermat Number" from the Notebook Archive (2002), https://notebookarchive.org/2019-07-0z4eh25

Download

