Other Free Encyclopedias » Science Encyclopedia » Science & Philosophy: Ambiguity - Ambiguity to Anticolonialism in Middle East - Ottoman Empire And The Mandate System

Amicable Numbers

pair divisors pairs proper

Two numbers are said to be amicable (i.e., friendly) if each one of them is equal to the sum of the proper divisors of the others, i.e., whole numbers less than the given numbers that divide the given number with no remainder. For example, 220 has proper divisors 1, 2, 4, 5, 10, 11, 20, 22, 44, 55, and 110. The sum of these divisors is 284. The proper divisors of 284 are 1, 2, 4, 71, and 142. Their sum is 220; so 220 and 284 are amicable. This is the smallest pair of amicable numbers.

The discovery of amicable numbers is attributed to the neo-Pythagorean philosopher Iamblichus (c.250-330), who credited Pythagoras with the original knowledge of their nature. The Pythagoreans believed that amicable numbers, like all special numbers, had a profound cosmic significance. A biblical reference (a gift of 220 goats from Jacob to Esau, Genesis 23: 14) is thought by some to indicate an earlier knowledge of amicable numbers.

No pairs of amicable numbers other than 220 and 284 were discovered by European mathematicians until 1636, when the French mathematician Pierre de Fermat (1601-1665) found the pair 18,496 and 17,296. A century later, the Swiss mathematician Leonhard Euler (1707-1783) made an extensive search and found about 60 additional pairs. Surprisingly, however, he overlooked the smallest pair, 1184 and 1210, which was subsequently discovered in 1866 by a 16-year-old boy, Nicolo Paganini.

During the medieval period, Arabian mathematicians preserved and developed the mathematical knowledge of the ancient Greeks. For example, The polymath Thabit ibn Qurra (836-901) formulated an ingenious rule for generating amicable number pairs: Let a = 3(2n) - 1, b = 3(2n-1) - 1, and c = 9(22n-1) - 1; then, if a, b, and c are primes, 2nab and 2nc are amicable. This rule produces 220 and 284 when n is 2. When n is 3, c is not a prime, and the resulting numbers are not amicable. For n = 4, it produces Fermat's pair, 17,296 and 18,416, skipping over Paganini's pair and others.

Amicable numbers serve no practical purpose, but professionals and amateurs alike have for centuries enjoyed seeking them and exploring their properties.

User Comments

Your email address will be altered so spam harvesting bots can't read it easily.
Hide my email completely instead?

Cancel or

Vote down Vote up

over 1 year ago


Packers and Movers in Pune go to:-
http://www.moveby5th.in/packers-and-movers-pune.html
Packers and Movers in Bangalore go to:-
http://www.moveby5th.in/packers-and-movers-bangalore.html
Packers and Movers in Hyderabad go to:-
http://www.moveby5th.in/packers-and-movers-hyderabad.html
Packers and Movers in Mumbai go to:-
http://www.moveby5th.in/packers-and-movers-mumbai.html
Packers and Movers in Gurgaon go to:-
http://www.moveby5th.in/packers-and-movers-gurgaon.html
Packers and Movers in Delhi go to:-
http://www.moveby5th.in/packers-and-movers-delhi.html

Vote down Vote up

13 days ago

https://pilcytotec.com/ obat-aborsi

https://pilcytotec.com/ obat-cytotec

https://pilcytotec.com/ cara menggugurkan-kandungan

http://aip.completeplanet.com/?thisPage=pilcytotec.com/obat-penggugur-kandungan

http://archive.org/search.php?query=pilcytotec.com/obat-penggugur-kandungan

http://astalavista.box.sk/cgi-bin/robot?srch=pilcytotec.com/obat-penggugur-kandungan

http://behold.cc/?textq=pilcytotec.com/obat-penggugur-kandungan

http://blekko.com/ws/pilcytotec.com/obat-penggugur-kandungan

http://duckduckgo.com/?q=pilcytotec.com/obat-penggugur-kandungan

http://ecosia.org/search.php?q=pilcytotec.com/obat-penggugur-kandungan

http://harvester42.fzk.de/jss/Search?searchTerm=pilcytotec.com/obat-penggugur-kandungan

http://iboogie.com/searchtree.asp?name_query=pilcytotec.com/obat-penggugur-kandungan

http://iwon.results.myway.com/AJmain.jhtml?searchfor=pilcytotec.com/obat-penggugur-kandungan

http://jexamples.com/fwd?action=srchRes&queryText=pilcytotec.com/obat-penggugur-kandungan

http://lookahead.surfwax.com/index-2011.html?term=pilcytotec.com/obat-penggugur-kandungan

http://m.yahoo.com/w/search%3B_ylt=?

submit=oneSearch&.intl=us&.lang=en&p=pilcytotec.com/obat-penggugur-kandungan

http://melzoo.com/en_GB/search?query=pilcytotec.com/obat-penggugur-kandungan

http://msxml.excite.com/search/web?q=pilcytotec.com/obat-penggugur-kandungan

http://newspaperarchive.com/tags/pilcytotec.com/obat-penggugur-kandungan

http://nova.rambler.ru/search?&query=pilcytotec.com/obat-penggugur-kandungan

http://omgili.com/pilcytotec.com/obat-penggugur-kandungan

Vote down Vote up

17 days ago

Impressive work! I really enjoyed reading this amazing stuff. I am bookmarking your page for more updates. http://waterdamagesanjoseca.com/mold-inspection-san-jose/ is the best services provider.

Vote down Vote up

26 days ago

http://obat-aborsi99.com/,, Obat Aborsi,,
http://obat-aborsi99.com/,, Jual Obat Penggugur Kandungan,,
http://obat-aborsi99.com/,, Obat Aborsi,,
http://obat-aborsi99.com/,, Jual Obat Penggugur Kandungan,,

Vote down Vote up

about 1 month ago

http://obat-aborsi99.com/uncategorized/jual-obat-aborsi-cytotec/
Jual Cytotec Obat Aborsi, Obat Penggugur Kandungan & Janin,
jual obat aborsi kandungan

Jual Obat Aborsi, Obat Penggugur Kandungan, Jual Obat Aborsi Tuntas, Jual Obat Aborsi Ampuh

Jual Obat Aborsi, Obat Penggugur Kandungan Ampuh

Vote down Vote up

3 months ago

Jual Cytotec Obat Aborsi Asli Obat Penggugur Kandungan, Obat Aborsi
Jual Obat Aborsi Murah, Obat Penggugur Kandungan menjual Obat Aborsi Aman, Obat Aborsi Asli Ampuh, Obat Cytotec, Obat Aborsi Manjur, dan Cara Menggugurkan Kandungan.
Obat Aborsi Cytotec dan Obat Gastrul
http://obat-penggugur-kandungan.biz/

http://aborsi-tuntas.com/
Jual Obat Aborsi, Klinik Aborsi, Jual Obat Cytotec, Obat Aborsi, Obat Penggugur Kandungan

Jual Cytotec Obat Aborsi Asli Obat Penggugur Kandungan
http://obat-aborsi99.com/ jual obat aborsi

Jual Obat Aborsi Alsi, Obat Penggugur Kandungan Aman, Jual Obat Aborsi Tuntas, Jual Obat Aborsi Ampuh
http://klinikfarma.com/

Jual Obat Aborsi, Obat Aborsi, Jual Obat Aborsi Asli Ampuh, Obat Penggugur Kandungan


Jual Obat Aborsi Cytotec, Obat Aborsi, Obat Aborsi Asli Ampuh

Jual Obat Aborsi, Obat Penggugur Kandungan
http://obat-penggugur.biz/

Obat Aborsi, Obat Penggugur kandungan
http://obat-penggugur-kandungan.com/

Jual Obat Aborsi, Obat Aborsi, Klinik Aborsi
http://klinik-aborsi.com

http://obat-aborsi-jakarta-bekasi.com/ jual Obat Aborsi jakarta, Obat Aborsi Bekasi Obat Aborsi

http://obat-aborsi-bandung-batam.com/ Obat Aborsi Bandung, Obat Aborsi Batam Obat Aborsi

Vote down Vote up

about 1 year ago

obat aborsi ampuh http://www.jualcytotecasli.com/ obat penggugur kandungan

Vote down Vote up

over 1 year ago

http://obataborsicytotectuntas.com/

OBAT ABORSI AMPUH

Vote down Vote up

over 1 year ago

Packers and Movers in Hyderabad go to:-
http://www.movers5th.in/packers-and-movers-hyderabad/
Packers and Movers in Bangalore go to:-
http://www.movers5th.in/packers-and-movers-bangalore/
Packers and Movers in Pune go to:-
http://www.movers5th.in/packers-and-movers-pune/

Vote down Vote up

over 1 year ago

Packers and Movers in Mumbai go to:-
http://www.movers5th.in/packers-and-movers-mumbai/
Packers and Movers in Delhi go to:-
http://www.movers5th.in/packers-and-movers-delhi/
Packers and Movers in Gurgaon go to:-
http://www.movers5th.in/packers-and-movers-gurgaon/

Vote down Vote up

over 1 year ago


Packers and Movers in Chennai go to:-
http://www.movers5th.in/packers-and-movers-chennai/
Packers and Movers in Noida go to:-
http://www.movers5th.in/packers-and-movers-noida/
Packers and Movers in Navi Mumbai go to:-
http://www.movers5th.in/packers-and-movers-navimumbai/
Packers and Movers in Thane go to:-
http://www.movers5th.in/packers-and-movers-thane/
Packers and Movers in Ghaziabad go to:-
http://www.movers5th.in/packers-and-movers-ghaziabad/
Packers and Movers in Faridabad go to:-
http://www.movers5th.in/packers-and-movers-faridabad/

Vote down Vote up

over 1 year ago

http://ivanzidni.pointblog.net/
http://ivanzidni.over-blog.com/
http://ivanzidni.hatenablog.com/
https://ivanzidniblog.wordpress.com/
http://ivanzidni.weebly.com/
http://ivanzidni.livejournal.com/
http://ivanzidni.tumblr.com/
http://ivan-zidni.blogspot.co.id/
http://ivanzidni.bravesites.com/
http://ivanzidni.blogdetik.com/
http://ivanzidni.mywapblog.com/

Vote down Vote up

almost 2 years ago


smppoker situs agen judi poker online uang asli terpercaya indonesia

http://defaonline.com/smppoker-situs-agen-judi-poker-online-uang-asli-terpercaya-indonesia/

cahayaqq.com agen poker dan domino online uang asli indonesia

http://defaonline.com/cahayaqq-com-agen-poker-dan-domino-online-uang-asli-indonesia/

qilinpoker.net agen poker domino qq ceme blackjack terpercaya

http://defaonline.com/qilinpoker-net-agen-poker-domino-qq-ceme-blackjack-terpercaya/

lenterapoker.com agen poker dan domino online terpercaya indonesia

http://defaonline.com/lenterapoker-com-agen-poker-dan-domino-online-terpercaya-indonesia/

CahayaPoker.net Agen Poker Online Bandar Kiu Terpercaya Indonesia

http://defaonline.com/cahayapoker-net-agen-poker-online-bandar-kiu-terpercaya-indonesia/

MejaQQ.com Agen Judi Poker DominoQQ BandarQ Online Terbesar di Asia

http://defaonline.com/mejaqq-com-agen-judi-poker-dominoqq-bandarq-online-terbesar-di-asia/

MejaQQ.com Agen Judi Poker DominoQQ BandarQ Online Terbesar di Asia

http://defaonline.com/cobaqq-com-agen-judi-poker-online-dan-bandar-domino-duit-asli-terpercaya/


niagapoker.com agen judi poker online dan bandar domino terpercaya

http://the-dsn.heck.in/niagapoker-com-agen-judi-poker-online.xhtml

http://agenjudipokeronlineindonesia.logdown.com

http://defaonlineindonesiaterpercaya.logdown.com/posts/431565-pokerwalet-com-agen-poker-online-dan-capsa-online-terbaik-terpercaya-di-indonesia

http://agenjudipokeronlineindonesia.logdown.com/posts/431602/cobaqq-com-agen-judi-poker-online-dan-bandar-domino-duit-asli-terpercaya

inulpoker.net agen poker online dan bandar ceme online terbaik terpercaya

http://agenjudipokeronlineindonesia.logdown.com/posts/431603/inulpoker-net-agen-poker-online-dan-bandar-ceme-online-terbaik-terpercaya

http://agenjudipokeronlineindonesia.logdown.com/posts/431629/cahayapoker-net-agen-poker-online-bandar-kiu-terpercaya-indonesia

http://agenjudipokeronlineindonesia.logdown.com/posts/445806/cemerlangpoker-com-agen-judi-poker-online-domino-online-uang-asli-terpercaya

http://defaonlineindonesiaterpercaya.logdown.com/posts/431565-pokerwalet-com-agen-poker-online-dan-capsa-online-terbaik-terpercaya-di-indonesia

http://defaonlineindonesiaterpercaya.logdown.com/posts/431533-cobaqq-com-agen-judi-poker-online-dan-bandar-domino-duit-asli-terpercaya

http://defaonlineindonesiaterpercaya.logdown.com

http://obataborsi45.com

https://www.searchenginejournal.com/50-social-bookmarking-sites-importance-of-user-generated-tags-votes-and-links/6066/

http://www.bibsonomy.org/user/defaonline

http://defaonlineblog.tumblr.com/

http://feraripoker.net

Vote down Vote up

about 2 years ago

The article says the amicable pair discovered by Fermat was 18,496 and 17,296. There's a type, it's actually 18,416 and 17,296

Vote down Vote up

about 7 years ago

its so much interesting articles....thanks to the author

Vote down Vote up

almost 10 years ago

Dear author,thanks a TON for writing this article and indirectly helping me do my similar research paper...though could you say who actually found out the numbers 220 and 284????I ve been searching for 4 hours and i still cant find it....:(

Vote down Vote up

almost 10 years ago

Maybe this is a litttle insufficient but I still think it is ok for contexts like this....

Vote down Vote up

about 10 years ago



dear creator,



thanks! youve helped a loton my research paper! however, i didnt leave my correct e mail adress because i dont know u. thanks again