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Difference between revisions of "Template:Generalized Fermat number"
m |
(a+b even -> no factor 2) |
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| Line 3: | Line 3: | ||
Template Generalized Fermat number | Template Generalized Fermat number | ||
| − | Collect the data of a [[Generalized Fermat number]] | + | 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. |
| + | |||
| + | If {{V|a}}+{{V|b}} is even, the numbers are always divisible by 2 and so this factor is not given here. | ||
==Prototype== | ==Prototype== | ||
| Line 32: | Line 34: | ||
|GFNDigits=1234 | |GFNDigits=1234 | ||
|GFNFactors= | |GFNFactors= | ||
| − | |||
3,5 | 3,5 | ||
3,209 | 3,209 | ||
| Line 52: | Line 53: | ||
|GFNDigits=1234 | |GFNDigits=1234 | ||
|GFNFactors= | |GFNFactors= | ||
| − | |||
3,5 | 3,5 | ||
3,209 | 3,209 | ||
| Line 63: | Line 63: | ||
|GFNRemarks=test | |GFNRemarks=test | ||
}} | }} | ||
| − | [[Category:Prime collections]]</noinclude><includeonly>__NOTOC__{{#ifeq:{{NAMESPACENUMBER}}|0|{{#vardefine:Ta|{{#explode:{{PAGENAME}}||-3}}}}{{#vardefine:Tb|{{#explode:{{PAGENAME}}||-2}}}}{{#vardefine:Tn|{{#explode:{{PAGENAME}}||-1}}}}{{#ifeq:{{#var:Tb}}|1|{{#ifeq:{{#var:Ta}}|2|{{DISPLAYTITLE:Fermat number 2<sup>2<sup>{{#var:Tn}}</sup></sup>+1}}|{{DISPLAYTITLE:Generalized Fermat number {{#var:Ta}}<sup>2<sup>{{#var:Tn}}</sup></sup>+1}}}}|{{DISPLAYTITLE:Generalized Fermat number {{#var:Ta}}<sup>2<sup>{{#var:Tn}}</sup></sup>+{{#var:Tb}}<sup>2<sup>{{#var:Tn}}</sup></sup>}}}}}} | + | [[Category:Prime collections]]</noinclude><includeonly>__NOTOC__{{#ifeq:{{NAMESPACENUMBER}}|0|{{#vardefine:Ta|{{#explode:{{PAGENAME}}||-3}}}}{{#vardefine:Tb|{{#explode:{{PAGENAME}}||-2}}}}{{#vardefine:Tn|{{#explode:{{PAGENAME}}||-1}}}}{{#ifeq:{{#var:Tb}}|1|{{#ifeq:{{#var:Ta}}|2|{{DISPLAYTITLE:Fermat number 2<sup>2<sup>{{#var:Tn}}</sup></sup>+1}}|{{DISPLAYTITLE:Generalized Fermat number {{#var:Ta}}<sup>2<sup>{{#var:Tn}}</sup></sup>+1}}}}|{{DISPLAYTITLE:Generalized Fermat number {{#var:Ta}}<sup>2<sup>{{#var:Tn}}</sup></sup>+{{#var:Tb}}<sup>2<sup>{{#var:Tn}}</sup></sup>{{#ifexpr:({{#var:Ta}}+{{#var:Tb}}) mod 2||/2}}}}}}}} |
==Current data== | ==Current data== | ||
{| class="primedata" | {| class="primedata" | ||
| Line 74: | Line 74: | ||
| align="right"| <b>{{Vn}} :</b> || {{{GFNn}}}--> | | align="right"| <b>{{Vn}} :</b> || {{{GFNn}}}--> | ||
|- | |- | ||
| − | {{!}} align="right"{{!}} <b>Number :</b> {{!}}{{!}} {{#ifeq:{{{GFNb}}}|1|{{#ifeq:{{{GFNa}}}|2|{{NGF|{{{GFNn}}}}}|{{NGF|{{{GFNa}}}|{{{GFNn}}}}}}}|{{NGF|{{{GFNa}}}|{{{GFNb}}}|{{{GFNn}}}}}}} {{#if:{{{GFNFDBid}}}|([http://factordb.com/index.php?id={{{GFNFDBid}}} FactorDB])}} | + | {{!}} align="right"{{!}} <b>Number :</b> {{!}}{{!}} {{#ifeq:{{{GFNb}}}|1|{{#ifeq:{{{GFNa}}}|2|{{NGF|{{{GFNn}}}}}|{{NGF|{{{GFNa}}}|{{{GFNn}}}}}}}|{{NGF|{{{GFNa}}}|{{{GFNb}}}|{{{GFNn}}}}}}} {{#ifexpr:({{{GFNa}}}+{{{GFNb}}}) mod 2|| div 2}} {{#if:{{{GFNFDBid}}}|([http://factordb.com/index.php?id={{{GFNFDBid}}} FactorDB])}} |
|- | |- | ||
{{#if:{{{GFNDigits|}}}| | {{#if:{{{GFNDigits|}}}| | ||
Revision as of 13:50, 28 July 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.
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=
}}
Parameters
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>
