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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]] (for {{Vb}}=1 this is a [[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=
−
7,14#1877#Ivan Pervushin;Édouard Lucas
+
3,5
−
17353230210429594579133099699123162989482444520899,15#2010-03-27#Michael Vang
+
3,209
 +
7,14
 +
15288227662166113,8
 +
N1100000002618011261,9
 +
P345
 
C1133,1100000000212123761
 
C1133,1100000000212123761
−
P345,1100000000212123762
+
U12345,4545
−
|GFNState=Composite
+
|GFNState=CF
 
|GFNRemarks=test
 
|GFNRemarks=test
 
}}
 
}}
Line 48: Line 58:
 
|GFNDigits=1234
 
|GFNDigits=1234
 
|GFNFactors=
 
|GFNFactors=
−
7,14#1877#Ivan Pervushin;Édouard Lucas
+
3,5
−
17353230210429594579133099699123162989482444520899,15#2010-03-27#Michael Vang
+
3,209
 +
7,14
 +
15288227662166113,8
 +
N1100000002618011261,9
 +
P345
 
C1133,1100000000212123761
 
C1133,1100000000212123761
−
P345,1100000000212123762
+
U12345,4545
−
|GFNState=Composite
+
|GFNState=CF
 
|GFNRemarks=test
 
|GFNRemarks=test
 
}}
 
}}
−
[[Category:Prime collections]]</noinclude><includeonly>__NOTOC__{{#ifeq:{{NAMESPACENUMBER}}|0|
+
[[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&nbsp;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||&nbsp;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||&nbsp;div 2}}}}}}}}
−
{{#ifeq:{{#explode:{{PAGENAME}}||-2}}|1|
 
−
  {{DISPLAYTITLE:Fermat Number 2<sup>2<sup>{{#explode:{{PAGENAME}}||-1}}</sup></sup>+1}}|
 
−
  {{DISPLAYTITLE:Generalized Fermat Number {{#explode:{{PAGENAME}}||-3}}<sup>2<sup>{{#explode:{{PAGENAME}}||-1}}</sup></sup> + {{#explode:{{PAGENAME}}||-2}}<sup>2<sup>{{#explode:{{PAGENAME}}||-1}}</sup></sup>}}}}
 
−
}}
 
 
==Current data==
 
==Current data==
 
{| class="primedata"
 
{| class="primedata"
 
| class="primetemplate color-PrimeTemplate" |
 
| class="primetemplate color-PrimeTemplate" |
 
  {|
 
  {|
−
   | align="right"| <b>{{V|a}} :</b> || {{{GFNa}}}
+
   | align="right"| <b>{{V|a}} {{Vb}} {{Vn}} :</b> || {{{GFNa}}} {{{GFNb}}} {{{GFNn}}}
 +
<!--  |-
 +
  | align="right"| <b>{{Vb}} :</b> || {{{GFNb}}}
 
   |-
 
   |-
−
   | align="right"| <b>{{V|b}} :</b> || {{{GFNb}}}
+
   | align="right"| <b>{{Vn}} :</b> || {{{GFNn}}}-->
 
   |-
 
   |-
−
  | align="right"| <b>{{Vn}} :</b> || {{{GFNn}}}
+
{{#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|{{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|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 77: Line 90:
 
   |-
 
   |-
 
{{#if:{{{GFNState|}}}|
 
{{#if:{{{GFNState|}}}|
−
   {{!}} align="right"{{!}} <b>State :</b> {{!}}{{!}} {{{GFNState}}}
+
   {{!}} align="right"{{!}} <b>State :</b> {{!}}{{!}} {{GFN state|{{{GFNState}}}}}
 
}}
 
}}
 
  |}
 
  |}
Line 87: Line 100:
 
}}
 
}}
 
==Factors==
 
==Factors==
−
{{#arraydefine:list|{{{GFNFactors}}}|/\n/}}{{#loop:i|0|{{#arraysize:list}}
+
{{#vardefine:facts|}}{{#arraydefine:GFNlist|{{{GFNFactors}}}|/\n/}}{{#loop:i|0|{{#arraysize:GFNlist}}
 
|<nowiki />
 
|<nowiki />
−
*{{#vardefine:k|{{#explode:{{#arrayindex:list|{{#var:i}}}}|,|0}}}}<!--
+
*{{#vardefine:k|{{#explode:{{#arrayindex:GFNlist|{{#var:i}}}}|,|0}}}}<!--
−
-->{{#vardefine:n|{{#explode:{{#explode:{{#arrayindex:list|{{#var:i}}}}|,|1}}|#|0}}}}<!--
+
-->{{#ifeq:{{Is Long number|{{#var:k}}}}|1|{{#vardefine:k|{{Long number:{{#var:k}}-NS}}({{Long number:{{#var:k}}-DI}})}}}}<!--
−
-->{{#vardefine:dat|{{#explode:{{#arrayindex:list|{{#var:i}}}}|#|1}}}}<!--
+
-->{{#vardefine:n|{{#explode:{{#explode:{{#arrayindex:GFNlist|{{#var:i}}}}|,|1}}|#|0}}}}<!--
 +
-->{{#vardefine:dat|}}<!--
 
-->{{#switch:{{#sub:{{#var:k}}|0|1}}
 
-->{{#switch:{{#sub:{{#var:k}}|0|1}}
−
|C=Composite {{#if:{{#var:n}}|[http://factordb.com/index.php?id={{#var:n}} {{#var:k}}]|{{#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 {{#if:{{#var:n}}|[http://factordb.com/index.php?id={{#var:n}} {{#var:k}}]|{{#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={{NPr|{{#var:k}}|{{#var:n}}}}
+
|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}}}}}}
 
}}<!--
 
}}<!--
−
-->{{#if:{{#var:dat}}|, found {{#var:dat}} by&#32;}}{{#arraydefine:disc|{{#explode:{{#arrayindex:list|{{#var:i}}}}|#|2}}|/;/}}{{#arrayprint:disc|,&#32;|@@@|[[@@@]]}}<!--
+
-->{{#ifexpr:{{#len:{{#var:dat}}}}>0|, found {{#explode:{{#var:dat}}|&#32;|0}} by&#32;}}{{Contribs|{{#explode:{{#var:dat}}|&#32;|1}}|}}
−
-->{{#ifeq:{{NAMESPACENUMBER}}|0|[[Category:Generalized Fermat number {{{GFNa}}} {{{GFNb}}}]]}}
+
}}{{#ifeq:{{NAMESPACENUMBER}}|0|[[Category:Generalized Fermat number {{{GFNa}}} {{{GFNb}}} Nums]]}}
−
}}</includeonly>
+
{{#if:{{#var:facts}}|
 +
==Factorization==
 +
<span class="longline">{{#vardefine:facts|{{#sub:{{#var:facts}}|2}}}}{{#var:facts}}}}</span>
 +
</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

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

Factors

Factorization

97 * 2468256835981809063232453773836025757474103798450369795022913537<64> * 114689 * 3913786281514524929<19> * 707275264749309881405141965802671548079179711820351316861777644606207216944972589404100097<90> * P<345> * C<1133> * U<12345>

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