M\u00fckemmel Say\u0131 : 6, 28, 496 gibi kendisi hari\u00e7 b\u00fct\u00fcn pozitif \u00e7arpanlar\u0131 toplam\u0131 kendisine e\u015fit olan say\u0131lara denir. M\u00fckemmel say\u0131lar sonsuz tane oldu\u011fu d\u00fc\u015f\u00fcn\u00fcl\u00fcr. Genel form\u00fclleri hen\u00fcz bulunamam\u0131\u015ft\u0131r. Ancak 2n(2n+1-1), say\u0131s\u0131n\u0131n her n \u00e7ift say\u0131s\u0131 ve 1 i\u00e7in m\u00fckemmel say\u0131 oldu\u011fu g\u00f6r\u00fclebilir. Tabi buradan m\u00fckemmel say\u0131lar\u0131n \u00e7ift say\u0131 olduklar\u0131 anlam\u0131 \u00e7\u0131kmamaktad\u0131r. Yani bu form\u00fcl\u00fcn t\u00fcm m\u00fckemmel say\u0131lar\u0131n ortak form\u00fcl\u00fc olup olmad\u0131\u011f\u0131 bilinmemektedir. Ancak \u015fu ana kadar bir tane tek m\u00fckemmel say\u0131 bulunamam\u0131\u015ft\u0131r…<\/p>\n
\u0130lk 11 m\u00fckemmel say\u0131 :<\/p>\n
6,
\n28,
\n496,
\n8128,
\n33550336,
\n8589869056,
\n137438691328
\n,2305843008139952128,
\n2658455991569831744654692615953842176,
\n19156194260823610729479337808430363813099732154816 9216 <\/p>\n
\u0130stedi\u011fimiz kadar say\u0131y\u0131 2’ye katlayarak toplayal\u0131m. Toplam asal say\u0131 oldu\u011funda, bu asal say\u0131y\u0131 son say\u0131yla \u00e7arpal\u0131m, \u00e7\u0131kan say\u0131 m\u00fckemmel say\u0131d\u0131r.
\nS\u00f6yleneni \u00f6rneklerle g\u00f6sterelim:1+2=3; 3 asal say\u0131; 3×2=6.; 6 m\u00fckemmel say\u0131. Ya da 1+2+4=7; 7 asal say\u0131; 7×4=28 m\u00fckemmel say\u0131. Veya 1+2+4+8+16=31 asal say\u0131; 31×16=496 m\u00fckemmel say\u0131.
\nGenel kural olarak; E\u011fer herhangi bir k>1 i\u00e7in 1+2+4+…+2k-1 =2k-1 asal ise; o zaman 2k-1(2k-1) bir m\u00fckemmel say\u0131d\u0131r. MS 100 civar\u0131nda, Nicomachus di\u011fer \u015feylerin yan\u0131nda, ispat gere\u011fi duymadan, m\u00fckemmel say\u0131larla ilgili \u015fu \u00f6zellikleri s\u0131ral\u0131yor:
\n1- N.ci. m\u00fckemmel say\u0131n\u0131n n basama\u011f\u0131 vard\u0131r.(1. Say\u0131 6, 2. say\u0131 28, 3.say\u0131 496, 4. say\u0131 8128) dikkat edelim ki hen\u00fcz 5. m\u00fckemmel say\u0131n\u0131n ka\u00e7 oldu\u011fu bilinmiyor. 2- B\u00fct\u00fcn m\u00fckemmel say\u0131lar \u00e7ifttir(sizin iddian\u0131z bu \u00f6zelli\u011fi yok ediyor) 3- B\u00fct\u00fcn m\u00fckemmel say\u0131lar, s\u0131ras\u0131yla 6 ve 8 ile biterler). 4- Herhangi bir k>1 i\u00e7in 2k-1 asal ise 2k-1(2k-1) bir m\u00fckemmel say\u0131d\u0131r ve m\u00fckemmel say\u0131lar\u0131n hepsini \u00fcreten bir algoritmad\u0131r. 5- Sonsuz say\u0131da m\u00fckemmel say\u0131 vard\u0131r.
\nTakip eden y\u00fczy\u0131llarda m\u00fckemmel say\u0131lar konusuna g\u00f6n\u00fcl veren bir\u00e7ok matematik\u00e7i oldu. Yaz\u0131l\u0131 kay\u0131tlarda 4.’den sonraki m\u00fckemmel say\u0131lara Arap matematik\u00e7i \u0130smail \u0130bn \u0130brahim \u0130bn Fallus’da(1194-1239) rastl\u0131yoruz. Verdi\u011fi 10 m\u00fckemmel say\u0131n\u0131n ilk 7 tanesi do\u011fru, 3 tanesi hatal\u0131. Nihayet 1536’da \u0130talyan matematik\u00e7i Pietro Cataldi, 211-1 say\u0131s\u0131n\u0131n asal olmad\u0131\u011f\u0131n\u0131(23.89=2047) g\u00f6sterdi. Bir asal say\u0131 olan 213-1=8191 ‘dan hareketle, 212(213-1)=33550336’n\u0131n bir m\u00fckemmel say\u0131 oldu\u011funu da buldu. 5. m\u00fckemmel say\u0131 8 basamakl\u0131yd\u0131. Nicomuchos’un iddialar\u0131ndan 1., 3., 4. zamanla \u00e7\u00fcr\u00fct\u00fcld\u00fcler. 6. say\u0131 1555’de J.Scheybl taraf\u0131ndan bulundu ise de 1977’ye kadar fark\u0131na var\u0131lmad\u0131\u011f\u0131ndan m\u00fckemmel say\u0131lar konusundaki geli\u015fmelere katk\u0131s\u0131 olmad\u0131.. 6. m\u00fckemmel say\u0131y\u0131 tekrar ve Scheybl den ba\u011f\u0131ms\u0131z olarak bulan gene Cataldi(1603) idi: 216(217-1)=8589869056. Bu s\u0131ra 8 de olmas\u0131na ra\u011fmen tekrar 6 ile biten bir m\u00fckemmel say\u0131yd\u0131. Cataldi 7. m\u00fckemmel say\u0131y\u0131 da bulan matematik\u00e7i oldu: 218(2191)=137438691328. M\u00fckemmel say\u0131larla ilgili \u00e7al\u0131\u015fan matematik\u00e7ilere Pierre de Fermat, Rene Descartes ve Marin Mersenne gibi \u00fcnl\u00fcleri de dahil edelim. Bu \u00e7al\u0131\u015fmalar s\u0131ras\u0131nda Mersenne Asallar\u0131’n\u0131n da bulundu\u011funu, Fermat’n\u0131n k\u00fc\u00e7\u00fck teoremi ad\u0131yla \u00fcnl\u00fc teoremin bu \u00e7al\u0131\u015fmalar\u0131n eseri oldu\u011funa de\u011findikten sonra, 8. m\u00fckemmel say\u0131y\u0131 bulan Euler’e gelelim: Euler, kendinden \u00f6nceki matematik\u00e7ilerden farkl\u0131 olarak, tek m\u00fckemmel say\u0131lar\u0131n da olabilece\u011fini ileri s\u00fcrd\u00fc. G\u00fcn\u00fcm\u00fcze kadar bu konuda yap\u0131lm\u0131\u015f olan \u00e7al\u0131\u015fmalar, ne bu iddian\u0131n do\u011frulu\u011funu ne de yanl\u0131\u015fl\u0131\u011f\u0131n\u0131 ispatlamaya yetmemi\u015ftir. G\u00fcn\u00fcm\u00fcze kadar 44 adet m\u00fckemmel say\u0131(hepsi \u00e7ift, hepsi 6 veya 8 ile bitiyor-ama s\u0131rayla de\u011fil) bulunmu\u015ftur. 44. m\u00fckemmel say\u0131n\u0131n 19 milyondan fazla basama\u011f\u0131 vard\u0131r. M\u00fckemmel say\u0131lar\u0131n tarihi k\u0131saca b\u00f6yle. 45.c\u0131 m\u00fckemmel ve ilk tek i\u00e7in say\u0131n\u0131z\u0131 bekliyoruz. Bu arada s\u00f6ylemeden ge\u00e7meyelim; Bat\u0131’da m\u00fckemmel say\u0131lara g\u00f6sterilen tutkunun gerisinde ilk say\u0131 olan 6’n\u0131n tanr\u0131n\u0131n d\u00fcnyay\u0131 6 g\u00fcnde yaratm\u0131\u015f olmas\u0131 inanc\u0131 ve Ay ay\u0131n\u0131n 2. say\u0131 kadar, yani 28 g\u00fcn olmas\u0131 da var.<\/p>\n","protected":false},"excerpt":{"rendered":"
M\u00fckemmel Say\u0131 : 6, 28, 496 gibi kendisi hari\u00e7 b\u00fct\u00fcn pozitif \u00e7arpanlar\u0131 toplam\u0131 kendisine e\u015fit olan say\u0131lara denir. M\u00fckemmel say\u0131lar sonsuz tane oldu\u011fu d\u00fc\u015f\u00fcn\u00fcl\u00fcr. Genel form\u00fclleri hen\u00fcz bulunamam\u0131\u015ft\u0131r. Ancak 2n(2n+1-1), say\u0131s\u0131n\u0131n her n \u00e7ift say\u0131s\u0131 ve 1 i\u00e7in m\u00fckemmel say\u0131 oldu\u011fu g\u00f6r\u00fclebilir. Tabi buradan m\u00fckemmel say\u0131lar\u0131n \u00e7ift say\u0131 olduklar\u0131 anlam\u0131 \u00e7\u0131kmamaktad\u0131r. Yani bu form\u00fcl\u00fcn t\u00fcm …<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[1404,1403],"tags":[7475],"class_list":["post-3231","post","type-post","status-publish","format-standard","hentry","category-matematik-odevleri","category-odevler","tag-mukemmel-sayi"],"_links":{"self":[{"href":"https:\/\/www.islamidavet.com\/kutuphane\/wp-json\/wp\/v2\/posts\/3231","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.islamidavet.com\/kutuphane\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.islamidavet.com\/kutuphane\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.islamidavet.com\/kutuphane\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.islamidavet.com\/kutuphane\/wp-json\/wp\/v2\/comments?post=3231"}],"version-history":[{"count":0,"href":"https:\/\/www.islamidavet.com\/kutuphane\/wp-json\/wp\/v2\/posts\/3231\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.islamidavet.com\/kutuphane\/wp-json\/wp\/v2\/media?parent=3231"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.islamidavet.com\/kutuphane\/wp-json\/wp\/v2\/categories?post=3231"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.islamidavet.com\/kutuphane\/wp-json\/wp\/v2\/tags?post=3231"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}