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Difference between revisions of "Template:Generalized Fermat number"
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(link to FactorDB ID/formula) |
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Collect the data of a [[Generalized Fermat number]] {{V|a}}<sup>2<sup>{{Vn}}</sup></sup> + {{V|b}}<sup>2<sup>{{Vn}}</sup></sup>, for {{Vb}}=1 this is a [[Fermat number]] {{V|a}}<sup>2<sup>{{Vn}}</sup></sup> + 1. | Collect the data of a [[Generalized Fermat number]] {{V|a}}<sup>2<sup>{{Vn}}</sup></sup> + {{V|b}}<sup>2<sup>{{Vn}}</sup></sup>, for {{Vb}}=1 this is a [[Fermat number]] {{V|a}}<sup>2<sup>{{Vn}}</sup></sup> + 1. | ||
| − | |||
| − | |||
==Prototype== | ==Prototype== | ||
| Line 22: | Line 20: | ||
==Parameters== | ==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 [[Factoring Database|FactorDB]] is created, otherwise only the number in standard notation is given. | ||
==See also== | ==See also== | ||
| + | *[[Template:GF Divisor|GF Divisor]] | ||
==Example== | ==Example== | ||
| Line 74: | Line 76: | ||
| align="right"| <b>{{Vn}} :</b> || {{{GFNn}}}--> | | align="right"| <b>{{Vn}} :</b> || {{{GFNn}}}--> | ||
|- | |- | ||
| − | {{!}} align="right"{{!}} <b> | + | {{#if:{{{GFNFDBid|}}}| |
| + | {{!}} align="right"{{!}} [[Factoring Database|FactorDB]] : {{!}}{{!}} <b>ID:</b> [http://factordb.com/index.php?id={{{GFNFDBid}}} {{{GFNFDBid}}}] / <b>Formula:</b> {{#ifeq:{{{GFNb}}}|1|{{#ifeq:{{{GFNa}}}|2|[http://factordb.com/index.php?query=2^2^{{{GFNn}}}%2B1 2<sup>2<sup>{{{GFNn}}}</sup></sup>+1]|[http://factordb.com/index.php?query={{#ifexpr:({{{GFNa}}}+{{{GFNb}}}) mod 2||(}}{{{GFNa}}}^2^{{{GFNn}}}%2B1{{#ifexpr:({{{GFNa}}}+{{{GFNb}}}) mod 2||)%2F2}} {{{GFNa}}}<sup>2<sup>{{{GFNn}}}</sup></sup>+1]}}| | ||
| + | [http://factordb.com/index.php?query={{#ifexpr:({{{GFNa}}}+{{{GFNb}}}) mod 2||(}}{{{GFNa}}}^2^{{{GFNn}}}%2B{{{GFNb}}}^2^{{{GFNn}}}{{#ifexpr:({{{GFNa}}}+{{{GFNb}}}) mod 2||)%2F2}} {{{GFNa}}}<sup>2<sup>{{{GFNn}}}</sup></sup>+{{{GFNb}}}<sup>2<sup>{{{GFNn}}}</sup></sup>]}} {{#ifexpr:({{{GFNa}}}+{{{GFNb}}}) mod 2|| div 2}}| | ||
| + | {{!}} align="right"{{!}} <b>Number :</b> {{!}}{{!}} {{#ifeq:{{{GFNb}}}|1|{{#ifeq:{{{GFNa}}}|2|2<sup>2<sup>{{{GFNn}}}</sup></sup>+1|{{{GFNa}}}<sup>2<sup>{{{GFNn}}}</sup></sup>+1}}|{{{GFNa}}}<sup>2<sup>{{{GFNn}}}</sup></sup>+{{{GFNb}}}<sup>2<sup>{{{GFNn}}}</sup></sup>}} {{#ifexpr:({{{GFNa}}}+{{{GFNb}}}) mod 2|| div 2}} | ||
| + | }} | ||
|- | |- | ||
{{#if:{{{GFNDigits|}}}| | {{#if:{{{GFNDigits|}}}| | ||
Revision as of 13:48, 4 August 2021
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.
See also
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
|
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>
