Template:Generalized Fermat number

Description

Template Generalized Fermat number

Collect the data of a Generalized Fermat number a²ⁿ + b²ⁿ, for b=1 this is a Fermat number a²ⁿ + 1.

Prototype

{{Generalized Fermat number
|GFNa=
|GFNb=
|GFNn=
|GFNFDBid=
|GFNDigits=
|GFNFactors=
|GFNState=
|GFNRemarks=
}}

Parameters

If 'GFNa'+'GFNb' is even, the numbers are always divisible by 2 and so 2 is not given as factor here.

If 'GFNFDBid' is given, a link to the FactorDB is created, otherwise only the number in standard notation is given.

For the case (a,b) = (8,1) the number for F_n(8,1) div 2²ⁿ+1 = 4²ⁿ − 2²ⁿ + 1 is given.

See also

• GF Divisor

Example

{{Generalized Fermat number
|GFNa=2
|GFNb=1
|GFNn=207
|GFNFDBid=1000000000002000017
|GFNDigits=1234
|GFNFactors=
3,5
3,209
7,14
15288227662166113,8
N1100000002618011261,9
P345
C1133,1100000000212123761
U12345,4545
|GFNState=CF
|GFNRemarks=test
}}

will create:

Current data

a b n : 2 1 207 FactorDB : ID: 1000000000002000017 / Formula: 22207+1 Digits : 1234 State : CF

Remarks :

test

Factors

• Proth 3•2⁵+1 (GF Divisor)
• Proth 3•2²⁰⁹+1 (GF Divisor), found 1956 by Raphael M. Robinson
• Proth 7•2¹⁴+1 (GF Divisor), found 1877 by Ivan Mikheevich Pervushin, Édouard Lucas
• Proth 15288227662166113•2⁸+1 (GF Divisor)
• Proth 138139700146349586211941790195...83(88)•2⁹+1 (GF Divisor)
• Prime P_<345>
• Composite C_<1133>
• Unknown U_<12345>

Factorization

97 * 2468256835981809063232453773836025757474103798450369795022913537_<64> * 114689 * 3913786281514524929_<19> * 707275264749309881405141965802671548079179711820351316861777644606207216944972589404100097_<90> * P_<345> * C_<1133> * U_<12345>