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'''Teorema Euklides–Euler''' adalah sebuah [[teorema]] dalam [[teori bilangan]] yang mengaitkan [[bilangan sempurna]] dengan [[bilangan prima Mersenne]]. Teorema ini mengatakan bahwa bilangan genap dikatakan sempurna [[jika dan hanya jika]] bilangan tersebut mempunyai bentuk {{math|2<sup>''p''−1</sup>(2<sup>''p''</sup> − 1)}}, dengan {{math|2<sup>''p''</sup> − 1}} adalah [[bilangan prima]]. Teorema ini dinamai dari matematikawan bernama [[Euklides]] yang membuktikan aspek dari teorema "jika", dan [[Leonhard Euler]] yang membuktikan aspek dari teorema "hanya jika".
{{Periksa terjemahan|en|Infinity symbol}}{{Infobox Symbols|mark=∞|name=Simbol takhingga|unicode={{unichar|221E|Infinity|html=}}|different from={{unichar|267E|Tanda kertas permanen |nlink= Kertas bebas asam| html=}}}}'''Simbol takhingga''' atau '''simbol ananta''' ('''{{Math|∞}}''') merupakan [[Daftar simbol matematika|simbol matematika]] yang mewakili konsep [[takhingga]]. Simbol ini disebut juga sebagai lemniskat,{{r|mind}} dinamai dari bentuk yang serupa dalam [[geometri aljabar]], yaitu kurva [[Kurva lemniskat|lemniskat]].{{r|beautiful}} Simbol ini juga disebut sebagai "angka delapan malas", yang berasal dari terminologi [[pencitraan merek ternak]].{{r|zero}}
Teorema ini telah diduga bahwa ada tak berhingga banyaknya bilangan prima Mersenne. Walaupun kebenaran dari konjektur ini masih belum terungkap, tetapi menurut teorema Euklides–Euler, ini menyerupai dengan sebuah konjektur yang katanya ada tak berhingga banyaknya bilangan sempurna genap. Sayangnya, masih dibelum ketahui adakah bilangan sempurna ganjil yang tunggal.<ref name="stillwell" />
Simbol takhingga pertama kali dipakai dalam matematika oleh [[John Wallis]] pada abad ke-17, walaupun simbol ini memiliki sejarah yang panjang dalam pemakaian lainnya. Dalam [[matematika]], simbol takhingga seringkali diartikan sebagai proses takhingga ([[takhingga potensial]]) daripada nilai takhingga ([[takhingga aktual]]). Simbol takhingga memiliki arti teknis lain yang berkaitan dengannya, seperti pemakaian kertas yang tahan lama dalam [[Penjilidan|penjilidan buku]]<u>,</u> dan dipakai sebagai nilai simbolis takhingga dalam kesusasteraan dan mistisisme modern. Dalam [[desain grafis]], simbol takhingga umumnya dipakai sebagai elemen logo; contohnya dalam badan logo dan desain lama seperti [[bendera Métis]].
== Pernyataan dan contoh ==
Simbol takhingga dan beberapa variasi lainnya tersedia di berbagai [[pengodean karakter]].
Sebuah billngan sempurna adalah sebuah [[bilangan asli]] yang sama dengan jumlah dari [[pembagi]] wajarnya, dan bilangan-bilangan tersebut lebih dari kecilnya dan kemudian membaginya sama rata (sampai tidak ada [[sisa]]). Sebagai contoh, pembagi wajar dari 6 adalah 1, 2, dan 3, yang hasilnya menjadi 6 saat dijumlahkan. Dengan demikian, 6 adalah bilangan sempurna.
Sebuah bilangan prima Mersenne adalah sebuah bilangan prima yang berbentuk {{math|1=''M''<sub>''p''</sub> = 2<sup>''p''</sup> − 1}}, sebuah bilangan yang lebih kecil dari [[Perpangkatan bilangan dua|perpangkatan dari dua]]. Agar bilangan dari bentuk tersebut berupa bilangan prima, maka {{mvar|p}} sendiri juga harus bilangan prima, tetapi tak semua bilangan prima menghasilkan bilangan prima Mersenne melalui cara ini. Sebagai contoh, {{nowrap|1=2<sup>3</sup> − 1 = 7}} adalah bilangan prima Mersenne, sedangkan {{nowrap|1=2<sup>11</sup> − 1 = 2047 = 23 × 89}} bukan.
== Sejarah ==
{{multiple image
| total_width = 480
| image1 = John Wallis by Sir Godfrey Kneller, Bt.jpg
| alt1 = Gambar John Wallis, dilukis oleh Sir Godfrey Kneller pada tahun 1701, dari National Portrait Gallery
| caption1 = [[John Wallis]] memperkenalkan simbol takhingga {{char|<math>\infty</math>}} dalam sastra matematika.
| image2 = Infinity symbol.svg
| alt2 = Ada delapan variasi simbol takhingga
| caption2 = Simbol <math>\infty</math> dalam berbagai [[rupa huruf]].
}}
Sejak zaman dulu, simbol lemniskat ini merupakan ragam hias yang umum. Contohnya, simbol ini umumnya dapat dilihat pada sisir [[zaman Viking]].{{r|viking}}<sup>[terj. masih kasar]</sup>
Teorema Eukildes–Euler mengatakan bahwa sebuah bilangan asli genap disebut sempurna jika dan hanya jika bilangan tersebut berbentuk {{math|2<sup>''p''−1</sup>''M''<sub>''p''</sub>}}, dengan {{math|''M''<sub>''p''</sub>}} adalah bilangan prima Mersenne.<ref name="stillwell">{{citation|title=Mathematics and Its History|series=[[Undergraduate Texts in Mathematics]]|first=John|last=Stillwell|authorlink=John Stillwell|publisher=Springer|year=2010|isbn=978-1-4419-6052-8|page=40|url=https://books.google.com/books?id=V7mxZqjs5yUC&pg=PA40}}.</ref> Sebagai contoh, bilangan sempurna 6 didapatkan ketika memasukkan {{math|1=''p'' = 2}} ke {{nowrap|1=2<sup>2−1</sup>{{mvar|M}}<sub>2</sub> = 2 × 3 = 6}}; dan memasukkan bilangan prima Mersenne 7 ke ekspresi yang serupa memperoleh bilangan sempurna 28.
[[John Wallis]], matematikawan asal Inggris, diakui bahwa ia telah memperkenalkan simbol takhingga beserta pengertiannya dalam matematika pada tahun 1655, dalam karyanya ''De sectionibus conicis''.{{r|sectionibus|wallis|notations}} Wallis tidak menjelaskan simbol yang ia pilih. Simbol yang dipilih diduga berupa bentuk bilangan Romawi yang berbeda, <u>tetapi bilangan yang dipilih masih belum jelas</u>. One theory proposes that the infinity symbol was based on the numeral for 100 million, which resembled the same symbol enclosed within a rectangular frame.{{r|beyond}} Adapula teori yang mengusul bahwa simbol tersebut berasal dari notasi CIↃ yang dipakai untuk mewakili 1000.{{r|unthinkable}} Instead of a Roman numeral, it may alternatively be derived from a variant {{nowrap|of {{char|ω}},}} the lower-case form of [[omega]], the last letter in the [[Greek alphabet]].{{r|unthinkable}}
== Sejarah ==
Karena keterbatasan tipografi dalam beberapa kasus, simbol-simbol lain yang menyerupai simbol takhingga dipakai dengan arti yang sama.{{r|notations}} [[Leonhard Euler]] menggunakan simbol huruf S yang terbalik dan menyamping yang menyerupai simbol lemniskat.{{r|notations-incorrect}} and even "O–O" has been used as a stand-in for the infinity symbol itself.{{r|notations}}
Euklides membuktikan bahwa {{math|2<sup>''p''−1</sup>(2<sup>''p''</sup> − 1)}} adalah sebuah bilangna prima genap dengan {{math|2<sup>''p''</sup> − 1}} adalah bilangan prima. Bukti tersebut adalah hasil terakhir tentang [[teori bilangan]] dalam buku miliknya, ''[[Elemen Euklides|Elements]]''; buku terakhir di ''Elements'' melibatkan [[bilangan irasional]], [[geometri padat]], dan [[rasio emas]]. Eukildes mengemukakan hasilnya dengan mengatakan bahwa jika [[deret geometrik]] terhingga dimulai dari 1 dengan rasio 2 mempunyai jumlah bilangan prima {{mvar|q}}, maka jumlah tersebut yang dikalikan dengan suku terakhir {{mvar|t}} di deret tersebut dikatakan sempurna. Ketika mengekspresikan bentuk tersebut, jumlah {{mvar|q}} dari deret terhingga menghasilkan bilangan prima Mersenne {{math|2<sup>''p''</sup> − 1}} dan suku terakhir {{mvar|t}} dalam deret tersebut merupakan perpangkatan dari dua {{math|2<sup>''p''−1</sup>}}. Euklides kemudian membuktikan bahwa {{math|''qt''}} dikatakan sempurna dengan mengamati deret geometrik dengan rasio 2 yang diawali dari {{mvar|q}}, dengan jumlah suku yang sama, sebanding dengan deret asli. Karena deret asli dijumlahkan sampai {{math|1=''q'' = 2''t'' − 1}}, maka deret kedua dijumlahkan sampai {{math|1=''q''(2''t'' − 1) = 2''qt'' − ''q''}}, dan kedua deret tersebut ditambahkan sampai {{math|2''qt''}}, dua kali dari bilangan sempurna sebelumnya. Akan tetapi, kedua deret tersebut terlepas dari satu sama lain serta (berdasarkan primalitas dari {{mvar|q}}) menghabiskan semua pembagi dari {{math|''qt''}}, sehingga {{math|''qt''}} mempunyai pembagi yang dijumlahkan sampai {{math|2''qt''}}, dan ini diperlihatkan bahwa bilangannya sempurna.<ref>{{citation|author=[[Euclid]]|title=The Thirteen Books of The Elements, Translated with introduction and commentary by Sir Thomas L. Heath, Vol. 2 (Books III–IX)|edition=2nd|publisher=Dover|year=1956|pages=421–426}}. See in particular Prop. IX.36.</ref>
== Pemakaian ==
=== Matematika ===
In mathematics, the infinity symbol is used more often to represent a [[potential infinity]],{{r|cosmic}} daripada kuantitas takhingga sebenarnya. Simbol takhingga dipakai diantaranya dalam [[bilangan real diperluas]], [[bilangan kardinal]], dan [[bilangan ordinal]] (notasi lain, seperti <math>\aleph_0</math> dan <math>\omega</math>, dipakai untuk bilangan takhingga). Misalnya, simbol takhingga pada bentuk [[penjumlahan]] dan [[limit]] dalam matematika seperti<math> \sum_{n=0}^{\infty} \frac{1}{2^n} = \lim_{x\to\infty}\frac{2^x-1}{2^{x-1}} = 2</math>,
biasanya diartikan bahwa variabel tersebut naik membesar mendekati takhingga<u>,</u> daripada diartikan sebagai nilai takhingga sebenarnya, meskipun hal tersebut dapat diartikan.<sup>[terj. masih kasar]</sup>{{r|convergence}}
Simbol takhingga juga dapat diartikan dalam [[titik di takhingga]], termasuk ketika hanya ada titik <u>yang sedang dipertimbangkan</u>.<sup>[terj, kasar]</sup> This usage includes, in particular, the infinite point of a [[projective line]],{{r|algebraic}} and the point added to a [[topological space]] to form its [[Alexandroff extension|one-point compactification]].{{r|hitchhiker}}
=== Other technical uses ===
[[Berkas:Infrarotindex_md_300_mm_IMGP1196.jpg|al=Minolta MD 4.5/300mm IF lens|jmpl|Side view of a camera lens, showing infinity symbol on the focal length indicator]]
In areas other than mathematics, the infinity symbol may take on other related meanings. For instance, it has been used in [[bookbinding]] to indicate that a book is printed on [[acid-free paper]] and will therefore be long-lasting.{{r|book}} On [[Camera|cameras]] and their [[Lens|lenses]], the infinity symbol indicates that the lens's [[focal length]] is [[Infinity focus|set to an infinite distance]], and is "probably one of the oldest symbols to be used on cameras".{{r|camera}}
=== Symbolism and literary uses ===
[[Berkas:RWS_Tarot_08_Strength.jpg|al=Strength tarot card, depicting a woman crowned by an infinity symbol, holding shut a lion's mouth|jmpl|The infinity symbol appears on several cards of the [[Rider–Waite tarot deck]].{{r|tarot}}]]
In modern mysticism, the infinity symbol has become identified with a variation of the [[ouroboros]], an ancient image of a snake eating its own tail that has also come to symbolize the infinite, and the ouroboros is sometimes drawn in figure-eight form to reflect this identification—rather than in its more traditional circular form.{{r|dreams}}
In the works of [[Vladimir Nabokov]], including ''[[The Gift (Nabokov novel)|The Gift]]'' and ''[[Pale Fire]]'', the figure-eight shape is used symbolically to refer to the [[Möbius strip]] and the infinite, as is the case in these books' descriptions of the shapes of bicycle tire tracks and of the outlines of half-remembered people. Nabokov's poem after which he entitled ''Pale Fire'' explicitly refers to "the miracle of the lemniscate".{{r|nabokov}} Other authors whose works use this shape with its symbolic meaning of the infinite include [[James Joyce]], in ''[[Ulysses (novel)|Ulysses]]'',{{r|sands}} and [[David Foster Wallace]], in ''[[Infinite Jest]]''.{{r|wallace}}
=== Graphic design ===
The well-known shape and meaning of the infinity symbol have made it a common [[Typography|typographic]] element of [[graphic design]]. For instance, the [[Métis flag]], used by the Canadian [[Métis]] people since the early 19th century, is based around this symbol.{{r|flags}} Different theories have been put forward for the meaning of the symbol on this flag, including the hope for an infinite future for Métis culture and its mix of European and [[First Nations in Canada|First Nations]] traditions,{{r|gaudry|dumont}} but also evoking the geometric shapes of Métic dances,{{r|racette}}, [[Celtic knot|Celtic knots]],{{r|prefontaine}} or [[Plains Indian Sign Language|Plains First Nations Sign Language]].{{r|barkwell}}
A [[rainbow]]-coloured infinity symbol is also used by the [[neurodiversity movement]], as a way to symbolize the infinite variation of the people in the movement and of human cognition.{{r|autism}} The [[Bakelite]] company took up this symbol in its corporate logo to refer to the wide range of varied applications of the synthetic material they produced.{{r|bakelite}} Versions of this symbol have been used in other trademarks, corporate logos, and emblems including those of [[Fujitsu]],{{r|brands}} [[Cell Press]],{{r|inspires}} and the [[2022 FIFA World Cup]].{{r|qatar}}
== Encoding ==
Simbol takhingga dikodekan sebagai {{unichar|221E|infinity}} dalam Unicode,{{r|compart}} dan <code>\infty</code>: <math>\infty</math> dalam [[LaTeX]].{{r|comprehensive}} Simbol takhingga yang dilingkari dipakai sebagai lambang [[kertas bebas asam]].
{{charmap|221E|map6=[[EUC-KR]]{{r|unicode-euc-kr}} / [[Unified Hangul Code|UHC]]{{r|unicode-uhc}}|namedref2=[[Common Locale Data Repository|CLDR]] text-to-speech name{{r|cldr}}|ref1char2=\acidfree|ref1char1=\infty|namedref1=[[LaTeX]]{{r|comprehensive}}|IncludeGB=1|map8char1=A1 DB|map8=[[Big5]]{{r|encoding}}|map7char1=A2 AC|map7=[[KPS 9566|EUC-KPS-9566]]{{r|unicode-kps}}|map6char1=A1 C4|map5char1=A1 E7|267E|map5=[[EUC-JP]]{{r|unicode-euc-jp}}|map4char1=81 87|map4=[[Shift JIS]]{{r|unicode-jis}}|map3char1=A5|map3=[[Symbol (typeface)#Encoding|Symbol Font encoding]]{{r|unicode-mac-symbol}}|map2char1=B0|map2=[[Mac OS Roman]]{{r|unicode-mac-roman}}|map1char1=EC|map1=[[Code page 437|OEM-437 (Alt Code)]]{{r|unicode-cp437}}|name2=permanent paper sign|name1=infinity|ref2char2=infinity}}
Ada kumpulan simbol Unicode yang juga diantaranya berupa bentuk simbol takhingga yang berbeda <u>that are less frequently available in fonts in the block [[Miscellaneous Mathematical Symbols-B]]</u>.{{r|unicode-misc}}
{{charmap|29DC|29DD|29DE|name1=incomplete infinity|name2=tie over infinity|name3=infinity negated with vertical bar|namedref1=[[LaTeX]]{{r|comprehensive}}|ref1char1=\iinfin|ref1char2=\tieinfty|ref1char3=\nvinfty}}
== See also ==
{{commons category|Infinity symbols}}
* [[Aleph number]]
* [[History of mathematical notation]]
* [[Lazy Eight (disambiguation)]]
== Rujukan ==
{{reflist|refs=<ref name=algebraic>{{cite book
| last = Perrin | first = Daniel
| isbn = 978-1-84800-056-8
| page = 28
| publisher = Springer
| title = Algebraic Geometry: An Introduction
| url = https://books.google.com/books?id=Vn1yR9qPvlMC&pg=PA28
| year = 2007}}</ref>
<ref name=autism>{{cite journal
| last = Gross | first = Liza
| date = September 2016
| doi = 10.1371/journal.pbio.2000958
| issue = 9
| journal = [[PLOS Biology]]
| page = e2000958
| title = In search of autism's roots
| volume = 14}}</ref>
<ref name=bakelite>{{cite journal
| last1 = Crespy | first1 = Daniel
| last2 = Bozonnet | first2 = Marianne
| last3 = Meier | first3 = Martin
| date = April 2008
| doi = 10.1002/anie.200704281
| issue = 18
| journal = Angewandte Chemie
| pages = 3322–3328
| title = 100 years of Bakelite, the material of a 1000 uses
| volume = 47}}</ref>
<ref name=barkwell>{{cite web |last1=Barkwell |first1=Lawrence J. |title=The Metis Infinity Flag |url=http://www.metismuseum.ca/resource.php/07245 |website=Virtual Museum of Métis History and Culture |publisher=Gabriel Dumont Institute |access-date=15 July 2020}}</ref>
<ref name=beautiful>{{cite book
| last = Erickson | first = Martin J.
| contribution = 1.1 Lemniscate
| contribution-url = https://books.google.com/books?id=LgeP62-ZxikC&pg=PA1
| isbn = 978-0-88385-576-8
| pages = 1–3
| publisher = [[Mathematical Association of America]]
| series = MAA Spectrum
| title = Beautiful Mathematics
| year = 2011}}</ref>
<ref name=beyond>{{cite book
| last = Maor | first = Eli | author-link = Eli Maor
| isbn = 0-691-02511-8
| mr = 1129467
| page = 7
| publisher = Princeton University Press | location = Princeton, New Jersey
| title = To Infinity and Beyond: A Cultural History of the Infinite
| url = https://books.google.com/books?id=pMY9DwAAQBAJ&pg=PA7
| year = 1991}}</ref>
<ref name=book>{{cite book
| last1 = Zboray | first1 = Ronald J.
| last2 = Zboray | first2 = Mary Saracino
| isbn = 978-0-8444-1015-9
| page = 49
| publisher = [[Center for the Book]], [[Library of Congress]]
| title = A Handbook for the Study of Book History in the United States
| url = https://archive.org/details/handbookforstudy0000zbor/page/49
| year = 2000}}</ref>
<ref name=brands>{{cite book
| last1 = Rivkin | first1 = Steve
| last2 = Sutherland | first2 = Fraser
| isbn = 978-0-19-988340-0
| page = 130
| publisher = Oxford University Press
| title = The Making of a Name: The Inside Story of the Brands We Buy
| url = https://books.google.com/books?id=WJVcgta-HToC&pg=PA130
| year = 2005}}</ref>
<ref name=camera>{{cite journal<!-- Although it would make more sense for this to be a conference, SAGE publishes it as a journal --> | last1 = Crist | first1 = Brian | last2 = Aurello | first2 = David N. | date = October 1990 | doi = 10.1177/154193129003400512 | issue = 5 | journal = Proceedings of the Human Factors Society Annual Meeting | pages = 489–493 | title = Development of camera symbols for consumers | volume = 34}}</ref>
<ref name=cldr>{{cite web |url=https://github.com/unicode-org/cldr/blob/master/common/annotations/en.xml |author=Unicode, Inc |author-link=Unicode Consortium |title=Annotations |work=[[Common Locale Data Repository]]|via=[[GitHub]]}}</ref>
<ref name=compart>{{Cite web|url=https://www.compart.com/en/unicode/U+221E|title=Unicode Character "∞" (U+221E)|work=Unicode|publisher=Compart AG|language=en|access-date=2019-11-15}}</ref>
<ref name=comprehensive>{{cite book|contribution=Table 294: stix Infinities|page=118|title=The Comprehensive LATEX Symbol List|first=Scott|last=Pakin|date=May 5, 2021|url=https://ctan.org/pkg/comprehensive|publisher=[[CTAN]]|access-date=2022-02-19}}</ref>
<ref name=convergence>{{cite journal
| last = Shipman | first = Barbara A.
| date = April 2013
| doi = 10.1080/10511970.2012.753963
| issue = 5
| journal = PRIMUS
| pages = 441–458
| title = Convergence and the Cauchy property of sequences in the setting of actual infinity
| volume = 23}}</ref>
<ref name=cosmic>{{cite book
| last = Barrow | first = John D. | author-link = John D. Barrow
| contribution = Infinity: Where God Divides by Zero
| contribution-url = https://books.google.com/books?id=uRg6iN10JCIC&pg=PA339
| isbn = 978-0-393-06177-2
| pages = 339–340
| publisher = W. W. Norton & Company
| title = Cosmic Imagery: Key Images in the History of Science
| year = 2008}}</ref>
<ref name=dreams>{{cite book|url=https://books.google.com/books?id=vhNNrX3bmo4C&pg=PA243|title=Dreams, Illusion, and Other Realities|last=O'Flaherty|first=Wendy Doniger|author-link=Wendy Doniger|publisher=University of Chicago Press|year=1986|isbn=978-0-226-61855-5|page=243}} The book also features this image on its cover.</ref>
<ref name=dumont>{{cite web |url= http://www.metisresourcecentre.mb.ca/index.php?option=com_content&view=article&id=2&Itemid=8 |title=The Métis flag |work=Gabriel Dumont Institute(Métis Culture & Heritage Resource Centre) |archive-url=https://web.archive.org/web/20130724192737/http://metisresourcecentre.mb.ca/index.php?option=com_content&view=article&id=2&Itemid=8 |archive-date=2013-07-24}}</ref>
<ref name=encoding>{{cite web |url=https://encoding.spec.whatwg.org/big5.html |title=big5 |work=Encoding Standard |publisher=[[WHATWG]] |last=van Kesteren |first=Anne |author-link=Anne van Kesteren}}</ref>
<ref name=flags>{{cite book|title=Native American Flags|first1=Donald T.|last1=Healy|first2=Peter J.|last2=Orenski|publisher=University of Oklahoma Press|year=2003|isbn=978-0-8061-3556-4|page=[https://archive.org/details/nativeamericanfl0000heal/page/284 284]|url=https://archive.org/details/nativeamericanfl0000heal/page/284}}</ref>
<ref name=gaudry>{{Cite journal|last=Gaudry |first=Adam |date=Spring 2018 |title= Communing with the Dead: The "New Métis," Métis Identity Appropriation, and the Displacement of Living Métis Culture |journal= American Indian Quarterly |volume=42 |issue=2 |pages=162–190 |jstor=10.5250/amerindiquar.42.2.0162 |doi=10.5250/amerindiquar.42.2.0162 |s2cid=165232342}}</ref>
<ref name=hitchhiker>{{cite book
| last1 = Aliprantis | first1 = Charalambos D. | author1-link = Charalambos D. Aliprantis
| last2 = Border | first2 = Kim C. | author2-link = Kim Border
| edition = 3rd
| isbn = 978-3-540-29587-7
| pages = 56–57
| publisher = Springer
| title = Infinite Dimensional Analysis: A Hitchhiker's Guide
| url = https://books.google.com/books?id=4vyXtR3vUhoC&pg=PA56
| year = 2006}}</ref>
<ref name=inspires>{{cite journal | last = Willmes | first = Claudia Gisela | date = January 2021 | doi = 10.1016/j.molmed.2020.11.001 | issue = 1 | journal = Trends in Molecular Medicine | page = 1 | pmid = 33308981 | title = Science that inspires | volume = 27}}</ref>
<ref name=mind>{{cite book
| last = Rucker | first = Rudy | author-link = Rudy Rucker
| isbn = 3-7643-3034-1
| location = Boston, Massachusetts
| mr = 658492
| page = 1
| publisher = Birkhäuser
| title = Infinity and the Mind: The science and philosophy of the infinite
| year = 1982}}</ref>
<ref name=nabokov>{{cite book|title=Nabokov: The Mystery of Literary Structures|first=Leona|last=Toker|publisher=Cornell University Press|year=1989|isbn=978-0-8014-2211-9|page=[https://archive.org/details/nabokovmysteryof00toke/page/159 159]|url=https://archive.org/details/nabokovmysteryof00toke|url-access=registration}}</ref>
<ref name=notations>{{Cite book|title-link=A History of Mathematical Notations|title=A History of Mathematical Notations, Volume II: Notations Mainly in Higher Mathematics|last=Cajori|first=Florian|author-link=Florian Cajori|publisher=Open Court |year=1929 |pages=44–48|contribution-url=https://archive.org/details/AHistoryOfMathematicalNotationVolII/page/n67|contribution=Signs for infinity and transfinite numbers}}</ref>
<ref name=notations-incorrect>{{harvtxt|Cajori|1929}} displays this symbol incorrectly, as a turned S without reflection. It can be seen as Euler used it on page 174 of {{cite journal|first=Leonhard|last=Euler|author-link=Leonhard Euler|language=la|title=Variae observationes circa series infinitas|journal=Commentarii academiae scientiarum Petropolitanae|volume=9|year=1744|pages=160–188|url=http://eulerarchive.maa.org/docs/originals/E072.pdf}}</ref>
<ref name=prefontaine>{{cite journal |last1=Darren R. |first1=Préfontaine |title=Flying the Flag, Editor's note. |journal=New Breed Magazine |date=2007 |issue=Winter 2007 |page=6 |url=http://www.metismuseum.ca/resource.php/05851 |access-date=26 August 2020}}</ref>
<ref name=qatar>{{cite news|url=https://www.aljazeera.com/news/2019/09/qatar-2022-football-world-cup-logo-unveiled-190903193444377.html|publisher=Al Jazeera|title=Qatar 2022: Football World Cup logo unveiled|date=September 3, 2019}}</ref>
<ref name=racette>{{Cite book |url= http://www.metismuseum.ca/media/document.php/12153.Flags%20of%20the%20Metis.pdf |title=Flags of the Métis |last=Racette |first=Calvin |publisher=Gabriel Dumont Institute |year=1987 |isbn=0-920915-18-3}}</ref>
<ref name=sands>{{cite book | last = Bahun | first = Sanja | editor1-last = Kim | editor1-first = Rina | editor2-last = Westall | editor2-first = Claire | contribution = 'These heavy sands are language tide and wind have silted here': Tidal voicing and the poetics of home in James Joyce's Ulysses | doi = 10.1057/9781137020758_4 | pages = 57–73 | publisher = Palgrave Macmillan | title = Cross-Gendered Literary Voices: Appropriating, Resisting, Embracing | year = 2012}}</ref>
<ref name=sectionibus>{{cite book|url=https://archive.org/details/bub_gb_03M_AAAAcAAJ|title=De Sectionibus Conicis, Nova Methodo Expositis, Tractatus|last=Wallis|first=John|year=1655|pages=[https://archive.org/details/bub_gb_03M_AAAAcAAJ/page/n17 4]|language=la|chapter=Pars Prima|author-link=John Wallis}}</ref>
<ref name=tarot>{{cite journal
| last = Armson | first = Morandir
| date = June 2011
| issue = 1
| journal = Literature & Aesthetics
| pages = 196–212
| title = The transitory tarot: an examination of tarot cards, the 21st century New Age and theosophical thought
| url = https://openjournals.library.sydney.edu.au/index.php/LA/article/viewFile/5056/5761
| volume = 21}} See in particular p. 203: "Reincarnation is symbolised in a number of cards within the Waite-Smith tarot deck. The primary symbols of reincarnation used are the infinity symbol or lemniscate, the wheel and the circle."</ref>
<ref name=unicode-cp437>{{cite web |url=https://unicode.org/Public/MAPPINGS/VENDORS/MICSFT/PC/CP437.TXT |title=cp437_DOSLatinUS to Unicode table |date=April 24, 1996 |last=Steele |first=Shawn |publisher=[[Unicode Consortium]] | access-date=2022-02-19}}</ref>
<ref name=unicode-euc-jp>{{cite web |url=https://raw.githubusercontent.com/unicode-org/icu/master/icu4c/source/data/mappings/euc-jp-2007.ucm |title=EUC-JP-2007 |publisher=[[Unicode Consortium]] |work=[[International Components for Unicode]] | access-date=2022-02-19|via=[[GitHub]]}}</ref>
<ref name=unicode-euc-kr>{{cite web |url=https://raw.githubusercontent.com/unicode-org/icu/master/icu4c/source/data/mappings/ibm-970_P110_P110-2006_U2.ucm |title=IBM-970 |publisher=[[Unicode Consortium]] |work=[[International Components for Unicode]]|date=May 9, 2007 | access-date=2022-02-19|via=[[GitHub]]}}</ref>
<ref name=unicode-jis>{{cite web |url=https://www.unicode.org/Public/MAPPINGS/OBSOLETE/EASTASIA/JIS/SHIFTJIS.TXT |title=Shift-JIS to Unicode |publisher=[[Unicode Consortium]] |date=December 2, 2015 | access-date=2022-02-19}}</ref>
<ref name=unicode-kps>{{cite web |url=https://unicode.org/Public/MAPPINGS/VENDORS/MISC/KPS9566.TXT |title=KPS 9566-2003 to Unicode |date=April 27, 2011 |publisher=[[Unicode Consortium]] | access-date=2022-02-19}}</ref>
<ref name=unicode-mac-roman>{{cite web |url=http://unicode.org/Public/MAPPINGS/VENDORS/APPLE/ROMAN.TXT |title=Map (external version) from Mac OS Roman character set to Unicode 2.1 and later |date=April 5, 2005 |publisher=[[Apple Inc.]] |via=[[Unicode Consortium]]| access-date=2022-02-19}}</ref>
Setelah Euklides membuktikannya selama bertahun-tahun, [[Alhazen]] menduga bahwa ''setiap'' bilangan sempurna genap merupakan bilangan berbentuk {{math|2<sup>''p''−1</sup>(2<sup>''p''</sup> − 1)}} dengan {{math|2<sup>''p''</sup> − 1}} bilangan prima, tetapi sayangnya ia belum daat membuktikan hasil tersebut.<ref>{{MacTutor Biography|id=Al-Haytham|title=Abu Ali al-Hasan ibn al-Haytham}}</ref> Hingga pada abad ke-18, lebih dari 2000 tahun setelah Euklides,<ref>{{citation
<ref name=unicode-mac-symbol>{{cite web |url=http://unicode.org/Public/MAPPINGS/VENDORS/APPLE/SYMBOL.TXT |title=Map (external version) from Mac OS Symbol character set to Unicode 4.0 and later|date=April 5, 2005 |publisher=[[Apple Inc.]] |via=[[Unicode Consortium]]| access-date=2022-02-19}}</ref>
| last1 = Pollack | first1 = Paul
| last2 = Shevelev | first2 = Vladimir
| doi = 10.1016/j.jnt.2012.06.008
| issue = 12
| journal = Journal of Number Theory
| mr = 2965207
| pages = 3037–3046
| title = On perfect and near-perfect numbers
| volume = 132
| year = 2012| arxiv = 1011.6160
| s2cid = 13607242
}}</ref> [[Leonhard Euler]] membuktikan bahwa rumus {{math|2<sup>''p''−1</sup>(2<sup>''p''</sup> − 1)}} akan menghasilkan bilangan sempurna genap.<ref name="stillwell" /><ref>{{citation|first=Leonhard|last=Euler|authorlink=Leonhard Euler|chapter=De numeris amicibilibus|trans-chapter=On amicable numbers|language=Latin|contribution-url=https://scholarlycommons.pacific.edu/euler-works/798/|title=Commentationes arithmeticae|volume=2|year=1849|pages=627–636}}. Originally read to the Berlin Academy on February 23, 1747, and published posthumously. See in particular section 8, p. 88.</ref> Jadi, bukti tersebut mempunyai kaitan antara bilangan sempurna genap dengan bilangan prima Mersenne, yang menyatakan masing-masing bilangan prima Mersenne menghasilkan sebuah bilangan sempurna genap, dan begitupula sebaliknya. Setelah Euler membuktikannya, banyak matematikawan lain telah menerbitkan bukti-bukti yang berbeda, di antaranya bukti [[Victor-Amédée Lebesgue]], [[Robert Daniel Carmichael]], [[Leonard Eugene Dickson]], John Knopfmacher, dan Wayne L. McDaniel. Bukti Dickson khususnya sudah umum dipakai dalam buku cetak.<ref>{{citation|last=Cohen|first=Graeme L.|date=March 1981|doi=10.2307/3617930|issue=431|journal=[[The Mathematical Gazette]]|jstor=3617930|pages=28–30|title=Even perfect numbers|volume=65}}</ref>
Teorema ini tercantum dalam sebuah situs yang memuat daftar dari "100 teorema matematika yang terkenal", dating from 1999, which later became used by Freek Wiedijk as a [[Benchmark (computing)|benchmark]] set to test the power of different [[proof assistant]]s. {{as of|2021}}, the proof of the Euclid–Euler theorem had been formalized in 5 of the 10 proof assistants recorded by Wiedijk.<ref>{{citation|first=Freek|last=Wiedijk|url=https://www.cs.ru.nl/~freek/100/|title=Formalizing 100 Theorems|publisher=Radboud University Institute for Computing and Information Sciences|access-date=2021-07-10}}</ref>
<ref name=unicode-misc>{{cite web|url=https://www.unicode.org/charts/PDF/U2980.pdf|title=Miscellaneous Mathematical Symbols-B|publisher=[[Unicode Consortium]]|archive-url=https://web.archive.org/web/20181112231107/https://www.unicode.org/charts/PDF/U2980.pdf|archive-date=2018-11-12|url-status=live| access-date=2022-02-19}}</ref>
== Proof ==
<ref name=unicode-uhc>{{cite web |url=https://www.unicode.org/Public/MAPPINGS/VENDORS/MICSFT/WINDOWS/CP949.TXT |title=cp949 to Unicode table |last=Steele |first=Shawn |publisher=[[Unicode Consortium]] |date=January 7, 2000 | access-date=2022-02-19}}</ref>
Euler's proof is short<ref name="stillwell" /> and depends on the fact that the [[Divisor function|sum of divisors]] function {{mvar|σ}} is [[multiplicative function|multiplicative]]; that is, if {{mvar|a}} and {{mvar|b}} are any two [[relatively prime]] integers, then {{math|1=''σ''(''ab'') = ''σ''(''a'')''σ''(''b'')}}. For this formula to be valid, the sum of divisors of a number must include the number itself, not just the proper divisors. A number is perfect if and only if its sum of divisors is twice its value.
=== Sufficiency ===
<ref name=unthinkable>{{cite book
One direction of the theorem (the part already proved by Euclid) immediately follows from the multiplicative property: every Mersenne prime gives rise to an even perfect number. When {{math|1=2<sup>''p''</sup> − 1}} is prime,
| last = Clegg | first = Brian | author-link = Brian Clegg (writer)
<math display-block>\sigma(2^{p-1}(2^p - 1)) = \sigma(2^{p-1})\sigma(2^p - 1).</math>
| contribution = Chapter 6: Labelling the infinite
The divisors of {{math|2<sup>''p''−1</sup>}} are {{math|1, 2, 4, 8, ..., 2<sup>''p''−1</sup>}}. The sum of these divisors is a [[geometric series]] whose sum is {{math|1=2<sup>''p''</sup> − 1}}. Next, since {{math|1=2<sup>''p''</sup> − 1}} is prime, its only divisors are {{math|1}} and itself, so the sum of its divisors is {{math|1=2<sup>''p''</sup>}}.
| isbn = 978-1-84119-650-3
| publisher = Constable & Robinson Ltd
| title = A Brief History of Infinity: The Quest to Think the Unthinkable
| year = 2003}}</ref>
Combining these,
<ref name=viking>{{cite journal
<math display=block>\begin{align}
| last = van Riel | first = Sjoerd
\sigma(2^{p-1}(2^p - 1)) &= \sigma(2^{p-1})\sigma(2^p - 1) \\
| journal = Lund Archaeological Review
&= (2^p - 1)(2^p) \\
| pages = 163–178
&= 2(2^{p-1})(2^p - 1).
| title = Viking Age Combs: Local Products or Objects of Trade?
\end{align}</math>
| url = https://journals.lub.lu.se/lar/article/view/21656
Therefore, {{math|2<sup>''p''−1</sup>(2<sup>''p''</sup> − 1)}} is perfect.<ref name="imsp">{{citation|title=Introduction to Mathematical Structures and Proofs|series=Undergraduate Texts in Mathematics|first=Larry|last=Gerstein|publisher=Springer|year=2012|isbn=978-1-4614-4265-3|at=Theorem 6.94, p. 339|url=https://books.google.com/books?id=qK9y768b1NQC&pg=PA339}}.</ref><ref name="pp">{{citation|title=A proof that all even perfect numbers are a power of two times a Mersenne prime|website=Prime Pages|url=https://primes.utm.edu/notes/proofs/EvenPerfect.html|access-date=2014-12-02|first=Chris K.|last=Caldwell}}.</ref><ref name="ntfagd">{{citation|title=Number Theory, Fourier Analysis and Geometric Discrepancy|volume=81|series=London Mathematical Society Student Texts|first=Giancarlo|last=Travaglini|publisher=Cambridge University Press|year=2014|isbn=978-1-107-04403-6|pages=26–27|url=https://books.google.com/books?id=mIaYAwAAQBAJ&pg=PA26}}.</ref>
| volume = 23
| year = 2017}} See p. 172: "Within this type the lemniscate (∞) is a commonly used motif."</ref>
=== Necessity ===
<ref name=wallace>{{cite book | last = Natalini | first = Roberto | editor1-last = Boswell | editor1-first = Marshall | editor2-last = Burn | editor2-first = Stephen J. | contribution = David Foster Wallace and the mathematics of infinity | doi = 10.1057/9781137078346_3 | pages = 43–57 | publisher = Palgrave Macmillan | series = American Literature Readings in the 21st Century | title = A Companion to David Foster Wallace Studies | year = 2013}}</ref>
In the other direction, suppose that an even perfect number has been given, and partially factor it as {{math|2<sup>''k''</sup>''x''}}, where {{mvar|x}} is odd. For {{math|2<sup>''k''</sup>''x''}} to be perfect, the sum of its divisors must be twice its value:
{{NumBlk|:|<math>2^{k+1}x = \sigma(2^k x) = (2^{k+1} - 1)\sigma(x).</math>|∗}}
The odd factor {{math|2<sup>''k''+1</sup> − 1}} on the right side of '''(∗)''' is at least 3, and it must divide {{mvar|''x''}}, the only odd factor on the left side, so {{math|1=''y'' = ''x''/(2<sup>''k''+1</sup> − 1)}} is a proper divisor of {{mvar|x}}. Dividing both sides of '''(∗)''' by the common factor {{math|1=2<sup>''k''+1</sup> − 1}} and taking into account the known divisors {{mvar|x}} and {{mvar|y}} of {{mvar|x}} gives
{{Block indent|left=1.6|<math>2^{k+1}y = \sigma(x) = x + y + {}</math>other divisors<math>{} = 2^{k+1}y + {}</math>other divisors.}}
For this equality to be true, there can be no other divisors. Therefore, {{mvar|y}} must be {{math|1}}, and {{mvar|x}} must be a prime of the form {{math|1=2<sup>''k''+1</sup> − 1}}.<ref name="imsp" /><ref name="pp" /><ref name="ntfagd" />
== References ==
<ref name=wallis>{{cite book
| last = Scott | first = Joseph Frederick
| edition = 2nd
| isbn = 0-8284-0314-7
| page = 24
| publisher = [[American Mathematical Society]]
| title = The mathematical work of John Wallis, D.D., F.R.S., (1616-1703)
| url = https://books.google.com/books?id=XX9PKytw8g8C&pg=PA24
| year = 1981}}</ref>
{{reflist}}
<ref name=zero>{{cite book|title=Zero to Lazy Eight: The Romance of Numbers|first1=Alexander|last1=Humez|first2= Nicholas D.|last2=Humez|first3= Joseph|last3= Maguire|publisher =Simon and Schuster|year=1993|isbn= 978-0-671-74281-2|page=18|url=https://books.google.com/books?id=X429EAr8g4kC&pg=PA18}}</ref>}}
| 