Visa källa för Prothtal

'''Prothtal''', uppkallat efter matematikern [[François Proth]], är inom [[talteori]]n ett [[tal]] av formen

:<math>k \cdot 2^n+1 </math>

där <math>k</math> är ett [[Jämna och udda tal|udda]] positivt [[heltal]] och <math>n</math> är ett positivt heltal sådant att <math>2^n > k</math>. Utan den sistnämnda termen skulle alla udda heltal större än 1 vara Prothtal.<ref>{{MathWorld |title=Proth Number |id=ProthNumber}}</ref>

De första Prothtalen är:

:[[3 (tal)|3]], [[5 (tal)|5]], [[9 (tal)|9]], [[13 (tal)|13]], [[17 (tal)|17]], [[25 (tal)|25]], [[33 (tal)|33]], [[41 (tal)|41]], [[49 (tal)|49]], [[57 (tal)|57]], [[65 (tal)|65]], [[81 (tal)|81]], [[97 (tal)|97]], [[113 (tal)|113]], [[129 (tal)|129]], [[145 (tal)|145]], [[161 (tal)|161]], [[177 (tal)|177]], [[193 (tal)|193]], [[209 (tal)|209]], [[225 (tal)|225]], [[241 (tal)|241]], [[257 (tal)|257]], [[289 (tal)|289]], [[321 (tal)|321]], [[353 (tal)|353]], [[385 (tal)|385]], [[417 (tal)|417]], [[449 (tal)|449]], [[481 (tal)|481]], [[513 (tal)|513]], [[545 (tal)|545]], [[577 (tal)|577]], [[609 (tal)|609]], [[641 (tal)|641]], [[673 (tal)|673]], [[705 (tal)|705]], [[737 (tal)|737]], [[769 (tal)|769]], [[801 (tal)|801]], [[833 (tal)|833]], [[865 (tal)|865]], [[897 (tal)|897]], [[929 (tal)|929]], [[961 (tal)|961]], [[993 (tal)|993]], [[1025 (tal)|1025]], [[1089 (tal)|1089]], [[1153 (tal)|1153]], [[1217 (tal)|1217]], [[1281 (tal)|1281]], [[1345 (tal)|1345]], [[1409 (tal)|1409]], … {{OEIS|A080075}}

[[Cullental]] (''n'' · 2<sup>''n''</sup> + 1) och [[Fermattal]] (2<sup>2''n''</sup> + 1) är specialfall av Prothtal.

== Prothprimtal ==
{{Huvudartikel|Prothprimtal}}

Ett [[Prothprimtal]] är ett Prothtal som även är [[primtal]].

De första Prothprimtalen är:

:[[3 (tal)|3]], [[5 (tal)|5]], [[13 (tal)|13]], [[17 (tal)|17]], [[41 (tal)|41]], [[97 (tal)|97]], [[113 (tal)|113]], [[193 (tal)|193]], [[241 (tal)|241]], [[257 (tal)|257]], [[353 (tal)|353]], [[449 (tal)|449]], [[577 (tal)|577]], [[641 (tal)|641]], [[673 (tal)|673]], [[769 (tal)|769]], [[929 (tal)|929]], [[1153 (tal)|1153]], [[1217 (tal)|1217]], [[1409 (tal)|1409]], [[1601 (tal)|1601]], [[2113 (tal)|2113]], [[2689 (tal)|2689]], [[2753 (tal)|2753]], [[3137 (tal)|3137]], [[3329 (tal)|3329]], [[3457 (tal)|3457]], [[4481 (tal)|4481]], [[4993 (tal)|4993]], [[6529 (tal)|6529]], [[7297 (tal)|7297]], [[7681 (tal)|7681]], [[7937 (tal)|7937]], [[9473 (tal)|9473]], [[9601 (tal)|9601]], [[9857 (tal)|9857]], [[10369 (tal)|10369]], [[10753 (tal)|10753]], [[11393 (tal)|11393]], [[11777 (tal)|11777]], [[12161 (tal)|12161]], [[12289 (tal)|12289]], [[13313 (tal)|13313]], … {{OEIS|A080076}}

Om ett Prothtal är ett primtal kan testas med [[Proths sats]] som säger att ett Prothtal <math>p</math> är primtal [[om och endast om]] det finns heltal <math>a</math> för vilka följande gäller:<ref>{{MathWorld |title=Proth's Theorem |id=ProthsTheorem}}</ref>

:<math>a^{\frac{p-1}{2}}\equiv -1\ \pmod{p}</math>

Det största kända Prothprimtalet (2010) är <math>19249 \cdot 2^{13018586} + 1</math>.<ref>Chris Caldwell, [http://primes.utm.edu/top20/page.php?id=66 The Top Twenty: Proth], from The [[Prime Pages]].</ref> Det hittades av [[Konstantin Agafonov]] och tillkännagavs den 5 maj 2007.<ref>[http://www.seventeenorbust.com/documents/press-050507.mhtml Press Release by Seventeen or Bust] {{Wayback|url=http://www.seventeenorbust.com/documents/press-050507.mhtml |date=20131219074325 }}. 5 May 2007.</ref> Det är också det största kända icke-[[Mersenneprimtal]]et.<ref>Chris Caldwell, [http://primes.utm.edu/top20/page.php?id=3 The Top Twenty: Largest Known Primes], from The [[Prime Pages]].</ref>

== Se även ==
* [[Sierpinskital]]
* [[PrimeGird]] – ett [[distributed computing]]-projekt som söker efter stora Prothprimtal

== Källor ==
* {{Enwp|url=//en.wikipedia.org/w/index.php?title=Proth_number&oldid=557555268|artikel=Proth number|datum=18 december 2013}}
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