已知最大質數：修订间差异

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質數，又名[[素数]]，是一個除1與自身之外沒有其他[[因數]]的[[正整数]]。[[欧几里得定理]]說明質數沒有上限，不少數學家與嗜好者故一直尋找大質數。
質數，又名[[素数]]，是一個除1與自身之外沒有其他[[因數]]的[[正整数]]。[[欧几里得定理]]說明質數沒有上限，不少數學家與嗜好者故一直尋找大質數。


不少大質數為[[梅森素数]]，定義為[[2的冪]]減去1的正整數。{{As of|2018|December}}，首八個已知大質數皆為梅森素数<ref>{{cite web |url=https://primes.utm.edu/primes/search.php?Number=100 |title=The largest known primes - Database Search Output |first1=Chris |last1=Caldwell |date= |publisher=Prime Pages |access-date=2018-06-03 |archive-date=2021-03-12 |archive-url=https://web.archive.org/web/20210312182058/https://primes.utm.edu/primes/search.php?Number=100 |dead-url=no }}</ref>。近十七次最大質數紀錄皆為梅森素数<ref name="computerhistory">{{cite web |url=http://primes.utm.edu/notes/by_year.html |title=The Largest Known Prime by Year: A Brief History |first1=Chris |last1=Caldwell |date= |publisher=Prime Pages |access-date=2016-01-20 |archive-date=2013-08-19 |archive-url=https://www.webcitation.org/6IzGFAMtA?url=http://primes.utm.edu/notes/by_year.html |dead-url=no }}</ref><ref>最後一個非梅森素数為[http://primes.utm.edu/primes/page.php?id=390 391,581 ⋅ 2<sup>216,193</sup> − 1] {{Wayback|url=http://primes.utm.edu/primes/page.php?id=390 |date=20210129154430 }}；參見[http://primes.utm.edu/notes/by_year.html The Largest Known Prime by Year: A Brief History] {{Wayback|url=http://primes.utm.edu/notes/by_year.html |date=20130808055216 }}，Caldwell着</ref>。所有梅森素数的[[二进制]]表示中，所有數字皆為1<ref>{{Cite web|url=http://www.personal.psu.edu/sxt104/class/Math140H/PerfectNum.html|title=Perfect Numbers|last=|first=|date=|website=Penn State University|archive-url=https://web.archive.org/web/20200803072640/http://www.personal.psu.edu/sxt104/class/Math140H/PerfectNum.html|archive-date=2020-08-03|access-date=2019-10-06|quote=An interesting side note is about the binary representations of those numbers...|dead-url=no}}</ref>。
不少大質數為[[梅森素数]]，定義為[[2的冪]]減去1的正整數。{{As of|2018|December}}，首八個已知大質數皆為梅森素数<ref>{{cite web |url=https://primes.utm.edu/primes/search.php?Number=100 |title=The largest known primes - Database Search Output |first1=Chris |last1=Caldwell |date= |publisher=Prime Pages |access-date=2018-06-03 |archive-date=2021-03-12 |archive-url=https://web.archive.org/web/20210312182058/https://primes.utm.edu/primes/search.php?Number=100 |dead-url=no }}</ref>。近十七次最大質數紀錄皆為梅森素数<ref name="computerhistory">{{cite web |url=http://primes.utm.edu/notes/by_year.html |title=The Largest Known Prime by Year: A Brief History |first1=Chris |last1=Caldwell |date= |publisher=Prime Pages |access-date=2016-01-20 |archive-date=2013-08-19 |archive-url=https://www.webcitation.org/6IzGFAMtA?url=http://primes.utm.edu/notes/by_year.html |dead-url=no }}</ref><ref>最後一個非梅森素数為[http://primes.utm.edu/primes/page.php?id=390 391,581 ⋅ 2<sup>216,193</sup> − 1] ；參見[http://primes.utm.edu/notes/by_year.html The Largest Known Prime by Year: A Brief History] ，Caldwell着</ref>。所有梅森素数的[[二进制]]表示中，所有數字皆為1<ref>{{Cite web|url=http://www.personal.psu.edu/sxt104/class/Math140H/PerfectNum.html|title=Perfect Numbers|last=|first=|date=|website=Penn State University|archive-url=https://web.archive.org/web/20200803072640/http://www.personal.psu.edu/sxt104/class/Math140H/PerfectNum.html|archive-date=2020-08-03|access-date=2019-10-06|quote=An interesting side note is about the binary representations of those numbers...|dead-url=no}}</ref>。


[[卢卡斯-莱默检验法]]的[[快速傅里叶变换]]比起其他方式能更快速尋找到[[梅森素数]]。
[[卢卡斯-莱默检验法]]的[[快速傅里叶变换]]比起其他方式能更快速尋找到[[梅森素数]]。
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[[电子前哨基金会]]亦為大質數的找尋設立了數個獎項<ref name="prizes"/>，互联网梅森素数大搜索亦有協調一億數位以上的質數搜索，並與成功尋找者分享电子前哨基金会所提供的150,000元美金獎金。
[[电子前哨基金会]]亦為大質數的找尋設立了數個獎項<ref name="prizes"/>，互联网梅森素数大搜索亦有協調一億數位以上的質數搜索，並與成功尋找者分享电子前哨基金会所提供的150,000元美金獎金。


1999年發現首個超過一百萬數位的質數，並取得50,000美元獎金<ref>Electronic Frontier Foundation, [https://www.eff.org/press/releases/big-prime-nets-big-prize Big Prime Nets Big Prize] {{Wayback|url=https://www.eff.org/press/releases/big-prime-nets-big-prize |date=20201023084104 }}.</ref>。2008年發現了超過一千萬數位的質數，並取得100,000美元獎金<ref name="prizes">{{cite web |url=https://www.eff.org/press/archives/2009/10/14-0 |title=Record 12-Million-Digit Prime Number Nets $100,000 Prize |date=2009-10-14 |work=Electronic Frontier Foundation |publisher=[[电子前哨基金会]] |accessdate=2011-11-26 |archive-date=2011-08-05 |archive-url=https://web.archive.org/web/20110805043609/http://www.eff.org/press/archives/2009/10/14-0 |dead-url=no }}</ref>。[[時代雜誌]]稱之為2008年第29名最佳發現<ref name="invention">{{cite news |url=http://www.time.com/time/specials/packages/article/0,28804,1852747_1854195_1854157,00.html |title=Best Inventions of 2008 - 29. The 46th Mersenne Prime |work=Time |publisher=[[时代公司]] |accessdate=2012-01-17 |date=2008-10-29 |archive-date=2013-08-22 |archive-url=https://web.archive.org/web/20130822215258/http://www.time.com/time/specials/packages/article/0,28804,1852747_1854195_1854157,00.html |dead-url=yes }}</ref>兩項獎金皆為互联网梅森素数大搜索的參加者。电子前哨基金会現為首個一億及十億數位的質數提供獎金<ref name="prizes"/>。
1999年發現首個超過一百萬數位的質數，並取得50,000美元獎金<ref>Electronic Frontier Foundation, [https://www.eff.org/press/releases/big-prime-nets-big-prize Big Prime Nets Big Prize] .</ref>。2008年發現了超過一千萬數位的質數，並取得100,000美元獎金<ref name="prizes">{{cite web |url=https://www.eff.org/press/archives/2009/10/14-0 |title=Record 12-Million-Digit Prime Number Nets $100,000 Prize |date=2009-10-14 |work=Electronic Frontier Foundation |publisher=[[电子前哨基金会]] |accessdate=2011-11-26 |archive-date=2011-08-05 |archive-url=https://web.archive.org/web/20110805043609/http://www.eff.org/press/archives/2009/10/14-0 |dead-url=no }}</ref>。[[時代雜誌]]稱之為2008年第29名最佳發現<ref name="invention">{{cite news |url=http://www.time.com/time/specials/packages/article/0,28804,1852747_1854195_1854157,00.html |title=Best Inventions of 2008 - 29. The 46th Mersenne Prime |work=Time |publisher=[[时代公司]] |accessdate=2012-01-17 |date=2008-10-29 |archive-date=2013-08-22 |archive-url=https://web.archive.org/web/20130822215258/http://www.time.com/time/specials/packages/article/0,28804,1852747_1854195_1854157,00.html |dead-url=yes }}</ref>兩項獎金皆為互联网梅森素数大搜索的參加者。电子前哨基金会現為首個一億及十億數位的質數提供獎金<ref name="prizes"/>。


== 已知最大質數歷史 ==
== 已知最大質數歷史 ==
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|style="text-align:right;"| 65,087
|style="text-align:right;"| 65,087
| 1989
| 1989
| 群組發現，包括約翰·布朗、{{tsl|en|Landon Curt Noll|藍登·克特·諾爾}}、B. K. 柏拉狄、哲恩·史密夫、喬爾·史密夫、沙治奧<ref>[https://www.jstor.org/stable/2324686 Letters to the Editor] {{Wayback|url=https://www.jstor.org/stable/2324686 |date=20210121232003 }}. ''The American Mathematical Monthly'' 97, no. 3 (1990), p. 214. Accessed May 22, 2020.</ref><ref>[https://primes.utm.edu/bios/code.php?code=Z Proof-code: Z] {{Wayback|url=https://primes.utm.edu/bios/code.php?code=Z |date=20210121072940 }}, The {{tsl|en|Prime Pages|}}.</ref>，為已知最大質數歷史中最大的非梅森素數。
| 群組發現，包括約翰·布朗、{{tsl|en|Landon Curt Noll|藍登·克特·諾爾}}、B. K. 柏拉狄、哲恩·史密夫、喬爾·史密夫、沙治奧<ref>[https://www.jstor.org/stable/2324686 Letters to the Editor] . ''The American Mathematical Monthly'' 97, no. 3 (1990), p. 214. Accessed May 22, 2020.</ref><ref>[https://primes.utm.edu/bios/code.php?code=Z Proof-code: Z] , The {{tsl|en|Prime Pages|}}.</ref>，為已知最大質數歷史中最大的非梅森素數。
|-
|-
| M<sub>756839</sub>
| M<sub>756839</sub>
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== 外部連結 ==
== 外部連結 ==
*[https://www.mersenne.org/primes/?press=M82589933 Press release about the largest known prime 2<sup>82,589,933</sup>−1] {{Wayback|url=https://www.mersenne.org/primes/?press=M82589933 |date=20181222180200 }}
* [https://www.mersenne.org/primes/?press=M82589933 Press release about the largest known prime 2<sup>82,589,933</sup>−1]
*[https://www.mersenne.org/primes/?press=M77232917 Press release about the former largest known prime 2<sup>77,232,917</sup>−1] {{Wayback|url=https://www.mersenne.org/primes/?press=M77232917 |date=20180104073001 }}
* [https://www.mersenne.org/primes/?press=M77232917 Press release about the former largest known prime 2<sup>77,232,917</sup>−1]
*[https://www.mersenne.org/primes/?press=M74207281 Press release about the former largest known prime 2<sup>74,207,281</sup>−1] {{Wayback|url=https://www.mersenne.org/primes/?press=M74207281 |date=20180107133230 }}
* [https://www.mersenne.org/primes/?press=M74207281 Press release about the former largest known prime 2<sup>74,207,281</sup>−1]


{{大數}}
{{大數}}
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