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Difference between revisions of "Template:Generalized Fermat number"
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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. |
==Prototype== | ==Prototype== | ||
| Line 20: | 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. | ||
| + | |||
| + | For the case ({{V|a}},{{V|b}}) = (8,1) the number for {{V|F}}<sub>{{Vn}}</sub>(8,1) div 2<sup>2<sup>{{Vn}}</sup></sup>+1 = 4<sup>2<sup>n</sup></sup> − 2<sup>2<sup>n</sup></sup> + 1 is given. | ||
==See also== | ==See also== | ||
| + | *[[Template:GF Divisor|GF Divisor]] | ||
==Example== | ==Example== | ||
| Line 32: | Line 38: | ||
|GFNDigits=1234 | |GFNDigits=1234 | ||
|GFNFactors= | |GFNFactors= | ||
| − | |||
3,5 | 3,5 | ||
3,209 | 3,209 | ||
7,14 | 7,14 | ||
15288227662166113,8 | 15288227662166113,8 | ||
| − | + | N1100000002618011261,9 | |
P345 | P345 | ||
C1133,1100000000212123761 | C1133,1100000000212123761 | ||
| + | U12345,4545 | ||
|GFNState=CF | |GFNState=CF | ||
|GFNRemarks=test | |GFNRemarks=test | ||
| Line 52: | Line 58: | ||
|GFNDigits=1234 | |GFNDigits=1234 | ||
|GFNFactors= | |GFNFactors= | ||
| − | |||
3,5 | 3,5 | ||
3,209 | 3,209 | ||
7,14 | 7,14 | ||
15288227662166113,8 | 15288227662166113,8 | ||
| − | + | N1100000002618011261,9 | |
P345 | P345 | ||
C1133,1100000000212123761 | C1133,1100000000212123761 | ||
| + | U12345,4545 | ||
|GFNState=CF | |GFNState=CF | ||
|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}}|{{#ifeq:{{#var:Ta}}|8|{{DISPLAYTITLE:Generalized Fermat number {{#var:Ta}}<sup>2<sup>{{#var:Tn}}</sup></sup>+1 div (2<sup>2<sup>{{#var:Tn}}</sup></sup>+1)}}|{{DISPLAYTITLE:Generalized Fermat number {{#var:Ta}}<sup>2<sup>{{#var:Tn}}</sup></sup>+1{{#ifexpr:({{#var:Ta}}+{{#var:Tb}}) mod 2|| div 2}}}}}}}}|{{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|| div 2}}}}}}}} |
==Current data== | ==Current data== | ||
{| class="primedata" | {| class="primedata" | ||
| Line 74: | Line 80: | ||
| 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]|{{#ifeq:{{{GFNa}}}|8|[http://factordb.com/index.php?query=({{{GFNa}}}^2^{{{GFNn}}}%2B1)/(2^2^{{{GFNn}}}%2B1) {{{GFNa}}}<sup>2<sup>{{{GFNn}}}</sup></sup>+1 div (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|}}}| | ||
| Line 94: | Line 103: | ||
|<nowiki /> | |<nowiki /> | ||
*{{#vardefine:k|{{#explode:{{#arrayindex:GFNlist|{{#var:i}}}}|,|0}}}}<!-- | *{{#vardefine:k|{{#explode:{{#arrayindex:GFNlist|{{#var:i}}}}|,|0}}}}<!-- | ||
| − | -->{{#ifeq:{{Is Long number|{{#var:k}}}}|1|{{#vardefine:k|{{Long number:{{#var:k}}-NS}}}}}}<!-- | + | -->{{#ifeq:{{Is Long number|{{#var:k}}}}|1|{{#vardefine:k|{{Long number:{{#var:k}}-NS}}({{Long number:{{#var:k}}-DI}})}}}}<!-- |
-->{{#vardefine:n|{{#explode:{{#explode:{{#arrayindex:GFNlist|{{#var:i}}}}|,|1}}|#|0}}}}<!-- | -->{{#vardefine:n|{{#explode:{{#explode:{{#arrayindex:GFNlist|{{#var:i}}}}|,|1}}|#|0}}}}<!-- | ||
-->{{#vardefine:dat|}}<!-- | -->{{#vardefine:dat|}}<!-- | ||
| Line 100: | Line 109: | ||
|C=Composite {{#vardefine:k|C<sub><{{#sub:{{#var:k}}|1}}></sub>}}{{#if:{{#var:n}}|[http://factordb.com/index.php?id={{#var:n}} {{#var:k}}]{{#vardefine:facts|{{#var:facts}} * [http://factordb.com/index.php?id={{#var:n}} {{#var:k}}]}}|{{#var:k}}{{#vardefine:facts|{{#var:facts}} * {{#var:k}}}}}} | |C=Composite {{#vardefine:k|C<sub><{{#sub:{{#var:k}}|1}}></sub>}}{{#if:{{#var:n}}|[http://factordb.com/index.php?id={{#var:n}} {{#var:k}}]{{#vardefine:facts|{{#var:facts}} * [http://factordb.com/index.php?id={{#var:n}} {{#var:k}}]}}|{{#var:k}}{{#vardefine:facts|{{#var:facts}} * {{#var:k}}}}}} | ||
|P=Prime {{#vardefine:k|P<sub><{{#sub:{{#var:k}}|1}}></sub>}}{{#if:{{#var:n}}|[http://factordb.com/index.php?id={{#var:n}} {{#var:k}}]{{#vardefine:facts|{{#var:facts}} * [http://factordb.com/index.php?id={{#var:n}} {{#var:k}}]}}|{{#var:k}}{{#vardefine:facts|{{#var:facts}} * {{#var:k}}}}}} | |P=Prime {{#vardefine:k|P<sub><{{#sub:{{#var:k}}|1}}></sub>}}{{#if:{{#var:n}}|[http://factordb.com/index.php?id={{#var:n}} {{#var:k}}]{{#vardefine:facts|{{#var:facts}} * [http://factordb.com/index.php?id={{#var:n}} {{#var:k}}]}}|{{#var:k}}{{#vardefine:facts|{{#var:facts}} * {{#var:k}}}}}} | ||
| − | |#default={{#if:{{#var:n}}|{{NPr|{{#var:k}}|{{#var:n}}}} ([[GF Divisor {{#var:k}} {{#var:n}}|GF Divisor]]){{#ifexist:GF Divisor {{#var:k}} {{#var:n}}|{{#vardefine:dat|{{GFDivisor get disc|GF Divisor {{#var:k}} {{#var:n}}|{{{GFNa}}},{{{GFNb}}},}}}}{{#vardefine:factn|{{GP|GF Divisor {{#var:k}} {{#var:n}}|GFNumber}}}}{{#if:{{#var:factn}}|{{#vardefine:facts|{{#var:facts}} * {{#ifeq:{{Is Long number|{{#var:factn}}}}|1|{{Long number:{{#var:factn}}-NL}}<sub><{{Long number:{{#var:factn}}-DI}}></sub>|{{#var:factn}}{{#ifexpr:{{#len:{{#var:factn}}}}>9|<sub><{{#len:{{#var:factn}}}}></sub>}}}}}}}}}}|{{#var:k}}{{#vardefine:facts|{{#var:facts}} * {{#var:k}}}}}} | + | |U=Unknown {{#vardefine:k|U<sub><{{#sub:{{#var:k}}|1}}></sub>}}{{#if:{{#var:n}}|[http://factordb.com/index.php?id={{#var:n}} {{#var:k}}]{{#vardefine:facts|{{#var:facts}} * [http://factordb.com/index.php?id={{#var:n}} {{#var:k}}]}}|{{#var:k}}{{#vardefine:facts|{{#var:facts}} * {{#var:k}}}}}} |
| + | |#default={{#if:{{#var:n}}|{{NPr|{{#var:k}}|{{#var:n}}}} ([[GF Divisor {{#var:k}} {{#var:n}}|GF Divisor]]){{#ifexist:GF Divisor {{#var:k}} {{#var:n}}|{{#vardefine:dat|{{GFDivisor get disc|GF Divisor {{#var:k}} {{#var:n}}|{{{GFNa}}},{{{GFNb}}},}}}}{{#vardefine:factn|{{GP|GF Divisor {{#var:k}} {{#var:n}}|GFNumber}}}}{{#ifeq:{{#sub:{{#var:factn}}|0|1}}|P|{{#vardefine:facts|{{#var:facts}} * P<sub><{{#sub:{{#var:factn}}|1}}></sub>}}|{{#if:{{#var:factn}}|{{#vardefine:facts|{{#var:facts}} * {{#ifeq:{{Is Long number|{{#var:factn}}}}|1|{{Long number:{{#var:factn}}-NL}}<sub><{{Long number:{{#var:factn}}-DI}}></sub>|{{#var:factn}}{{#ifexpr:{{#len:{{#var:factn}}}}>9|<sub><{{#len:{{#var:factn}}}}></sub>}}}}}}}}}}|{{#var:k}}{{#vardefine:facts|{{#var:facts}} * {{#var:k}}}}}}|{{#var:k}}{{#vardefine:facts|{{#var:facts}} * {{#var:k}}}}}} | ||
}}<!-- | }}<!-- | ||
-->{{#ifexpr:{{#len:{{#var:dat}}}}>0|, found {{#explode:{{#var:dat}}| |0}} by }}{{Contribs|{{#explode:{{#var:dat}}| |1}}|}} | -->{{#ifexpr:{{#len:{{#var:dat}}}}>0|, found {{#explode:{{#var:dat}}| |0}} by }}{{Contribs|{{#explode:{{#var:dat}}| |1}}|}} | ||
| − | }}{{#ifeq:{{NAMESPACENUMBER}}|0|[[Category:Generalized Fermat number {{{GFNa}}} {{{GFNb}}} | + | }}{{#ifeq:{{NAMESPACENUMBER}}|0|[[Category:Generalized Fermat number {{{GFNa}}} {{{GFNb}}} Nums]]}} |
{{#if:{{#var:facts}}| | {{#if:{{#var:facts}}| | ||
==Factorization== | ==Factorization== | ||
| − | {{#vardefine:facts|{{#sub:{{#var:facts}}|2}}}}{{#var:facts}}}} | + | <span class="longline">{{#vardefine:facts|{{#sub:{{#var:facts}}|2}}}}{{#var:facts}}}}</span> |
</includeonly> | </includeonly> | ||
Latest revision as of 08:51, 16 September 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.
For the case (a,b) = (8,1) the number for Fn(8,1) div 22n+1 = 42n − 22n + 1 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
N1100000002618011261,9
P345
C1133,1100000000212123761
U12345,4545
|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 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>
