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.

If a+b is even, the numbers are always divisible by 2 and so this factor is not given here.

Prototype

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

Example

{{Generalized Fermat number
|GFNa=2
|GFNb=1
|GFNn=207
|GFNFDBid=1000000000002000017
|GFNDigits=1234
|GFNFactors=
3,5
3,209
7,14
15288227662166113,8
N1100000000000108402,7
P345
C1133,1100000000212123761
|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 prime 2 Long number:N1100000000000108402-NS(Long number:N1100000000000108402-DI)|Proth Long number:N1100000000000108402-NS(Long number:N1100000000000108402-DI)•2⁷+1]] ([[GF Divisor Long number:N1100000000000108402-NS(Long number:N1100000000000108402-DI) 7|GF Divisor]])Long number:N1100000000000108402-NS(Long number:N1100000000000108402-DI)
• Prime P_<345>
• Composite C_<1133>

Factorization

97 * 2468256835981809063232453773836025757474103798450369795022913537_<64> * 114689 * 3913786281514524929_<19> * Long number:N1100000000000108402-NS(Long number:N1100000000000108402-DI) * P_<345> * C_<1133>