Teorema de Erdős–Kac

Teorema de Erdős–Kac em teoria dos números, assim nomeado por ter sido provado pelos matemáticos Paul Erdős e Mark Kac,[1] também conhecido como o teorema fundamental da teoria probabilística dos números, diz que se ω(n) é o número de fatores primos distintos de n, então, dizendo livremente, a distribuição de probabilidade de

é a distribuição normal padrão. Esta é um extensão mais aprofundada das ideias do Teorema de Hardy–Ramanujan, que diz que a ordem da função aritmética ω(n) é log log n com um típico erro de tamanho .[2]

Mais precisamente, para algum número a < b,

onde é a distribuição normal (ou "Gaussiana"), definida como

Enunciando de modo heurístico, Erdős e Kac provaram que se n é um inteiro suficientemente grande escolhido aleatoriamente, então o número de fatores primos distintos de n tem aproximadamente a distribuição normal com média e variancia log log n.

Isto significa que a construção de um número em torno de um bilhão requer em média uma base de três fatores primos.[3]

Por exemplo 1,000,000,003 = 23 × 307 × 141623.

n Número de

dígitos de n

Número médio

de primos distintos

desvio

padrão

1,000 4 2 1.4
1,000,000,000 10 3 1.7
1,000,000,000,000,000,000,000,000 25 4 2
1065 66 5 2.2
109,566 9,567 10 3.2
10210,704,568 210,704,569 20 4.5
101022 1022+1 50 7.1
101044 1044+1 100 10
1010434 10434+1 1000 31.6
Ficheiro:EKT plot.svg
Uma distribuição gaussiana de fatores primos distintos ilustrando o Teorema de Erdos-Kac

Em torno de 12.6% dos números de 10000 dígitos são construídos numa base de 10 fatores primos distintos e em torno de 68% (±σ) são construídos numa base entre 7 e 13 fatores primos distintos.

Se uma esfera oca com o tamanho do planeta Terra fosse preenchida com areia fina, teria por volta de 1033 grãos. Um volume do tamanho do Universo observável teria em torno de 1093 grãos de areia. Não pode haver espaço para 10185 cordas quânticas em tal universo.

Números desta magnitude — com 186 dígitos — requerem em média 6 fatores primos para construção.

Ver também

Referências

  1. ↑ Erdős, Paul; Kac, Mark (1940). «The Gaussian Law of errors in the Theory of Additive Number Theoretic Function». American Journal of Mathematics. 62 (1/4): 738–742. ISSN 0002-9327. Zbl 0024.10203 
  2. ↑ Kuo, Wentang; Liu, Yu-Ru (2008). «The Erdős–Kac theorem and its generalizations». In: De Koninck, Jean-Marie; Granville, Andrew; Luca, Florian. Anatomy of integers. Based on the CRM workshop, Montreal, Canada, March 13--17, 2006. Col: CRM Proceedings and Lecture Notes. 46. Providence, RI: American Mathematical Society. pp. 209–216. ISBN 978-0-8218-4406-9. Zbl 1187.11024 
  3. ↑ Kac, Mark (1959). Statistical Independence in Probability, Analysis and Number Theory. [S.l.]: John Wiley and Sons, Inc. 

Ligações externas

Read other articles:

Kema, Minahasa Utara

KemaKecamatanPeta lokasi Kecamatan KemaNegara IndonesiaProvinsiSulawesi UtaraKabupatenMinahasa UtaraKode Kemendagri71.06.01 Kode BPS7106010 Desa/kelurahan10 Perahu di pantai Kema (tahun 1930-an) Kema adalah sebuah kecamatan di Kabupaten Minahasa Utara, Sulawesi Utara, Indonesia. Batas wilayah Kecamatan Kema adalah di sebelah utara berbatasan dengan Kota Bitung, di sebelah timur dengan Laut Maluku, di sebelah selatan dengan Kecamatan Kombi, Kabupaten Minahasa, dan di sebelah barat dengan ...

 

Raden Pamanah Rasa (seri televisi)

Untuk tokoh yang dijuluki Raden Pamanah Rasa hingga menjadi salah satu Raja Sunda Galuh, lihat Sri Baduga Maharaja. Raden Pamanah RasaGenre Drama Epos Laga Fantasi PembuatMNC PicturesBerdasarkanRaden Pamanah Rasaoleh Enang Rokajat AsuraSkenario Ichal Beng(Eps. 1—6) Rachmawati(Eps. 7—9) CeritaEnang Rokajat AsuraSutradaraDedy MercySutradara lagaAhmad PotabugaPemeran Ferdi Ali Bima Azriel Kirana Larasati Fatmasury Penggubah lagu temaAhmad DhaniLagu pembukaIman oleh Ahmad DhaniLagu penutupIma...

 

Northern Maramureș

Northern Maramureș (gold) as part of the Zakarpattia Oblast of Ukraine, with district boundaries shown Northern Maramureș (Romanian: Maramureșul de Nord, [maraˈmureʃul de ˈnord]; Hungarian: Észak-Máramaros; Ukrainian: Північна Мараморщина, romanized: Pivnichna Maramorshchyna) is a geographic-historical region comprising roughly the eastern half of the Zakarpattia Oblast in southwestern Ukraine, near the border with Romania. Until 1920, it was part of t...

Tetrahydropalmatine

Isoquinoline alkaloid, found mainly in Corydalis TetrahydropalmatineIdentifiers IUPAC name (13aS)-2,3,9,10-tetramethoxy-6,8,13,13a-tetrahydro-5H-isoquinolino[2,1-b]isoquinoline CAS Number483-14-7 NPubChem CID72301ChemSpider65252 YUNII3X69CO5I79KEGGC02890 YChEBICHEBI:16563 YChEMBLChEMBL487182 YECHA InfoCard100.241.370 Chemical and physical dataFormulaC21H25NO4Molar mass355.434 g·mol−13D model (JSmol)Interactive image SMILES O(c1c4c(ccc1OC)C[C@H]3c2c(cc(OC)c(OC)...

 

Ikan laut dalam

Ikan sungut ganda Ikan laut dalam adalah istilah kolektif untuk ikan yang hidup dalam kegelapan di bawah permukaan perairan yang disinari matahari, yaitu di bawah epipelagik atau zona fotik di lautan. Sejauh ini ikan lentera adalah ikan laut dalam yang paling banyak. Ikan laut lainnya termasuk Ikan lentera, ikan senter, hiu pemotong,ikan mulut sikat, ikan sungut ganda, ikan beludak dan beberapa spesies dari puital. Hanya sekitar 2% spesies laut yang diketahui menghuni lingkungan pelagik, diba...

 

Kh-31

Kh-31 (Rusia: Х-31; AS-17 'Krypton')[1] adalah rudal udara-ke-permukaan Rusia yang dibawa oleh pesawat seperti MiG-29 atau Su-27. Ini adalah rudal sea skimming dengan jangkauan 110 kilometer (60 nm, 70 mil) atau lebih dan mampu Mach 3,5, dan adalah rudal anti-kapal supersonik pertama yang bisa diluncurkan oleh pesawat taktis.[2] Ada beberapa varian, yang terbaik adalah dikenal sebagai rudal anti-radiasi (ARM) tetapi ada juga anti-pengiriman dan versi sasaran drone. Telah ada...

森川智之

森川智之配音演员本名同上原文名森川 智之（もりかわ としゆき）罗马拼音Morikawa Toshiyuki昵称モリモリ[1]、帝王[1]国籍 日本出生 (1967-01-26) 1967年1月26日（57歲） 日本東京都品川區[1]（神奈川縣川崎市[2]、橫濱市[3]成長）职业配音員、旁白、歌手、藝人音乐类型J-POP出道作品外國人取向的日語教材代表作品但丁（Devil May Cry）D-boy（宇宙騎...

 

Мамонтовка (Пушкино)

У этого термина существуют и другие значения, см. Мамонтовка. Мамонтовка Территориальная администрация пос. Мамонтовка 56°00′ с. ш. 37°49′ в. д.HGЯO Страна  Россия[1] Город Пушкино Прежний статус посёлок городского типа Год включения в черту города 2003  Медиа...

 

دوران الأرض

دوران الأرضتعديل - تعديل مصدري - تعديل ويكي بيانات دوران الأرض هو حركة دوران الكرة الأرضية المجسمة حول محورها، حيث تدور الأرض باتجاه الشرق. وعندما ينظر إليها من جهة القطب الشمالي فإن الأرض تدور بعكس دوران عقارب الساعة (وستبدو عكس ذلك في حال نظر إليها من جهة القطب الجنوبي). ال...

Luis Muñoz de Guzmán

Luis Muñoz de Guzmán Retrato en óleo de Luis Muñoz de Guzmán. Gobernador del Reino de ChilePresidente de la Real Audiencia 30 de enero de 1802-11 de febrero de 1808Monarca Fernando VIIPredecesor Francisco Tadeo Díez de MedinaSucesor Juan Rodríguez Ballesteros 26.º Presidente de la Real Audiencia de Quito 1791-1797Monarca Carlos IVPredecesor Antonio Mon y VelardeSucesor Francisco de CarondeletVirrey José Manuel de Ezpeleta Información personalNacimiento 1735 Sevilla, Sevilla, Reino d...

 

2011 New England Patriots season

52nd season in franchise history; fourth Super Bowl loss 2011 New England Patriots seasonOwnerRobert KraftHead coachBill BelichickHome fieldGillette StadiumResultsRecord13–3Division place1st AFC EastPlayoff finishWon Divisional Playoffs(vs. Broncos) 45–10Won AFC Championship(vs. Ravens) 23–20 Lost Super Bowl XLVI(vs. Giants) 17–21Pro BowlersQB Tom BradyWR Wes WelkerTE Rob GronkowskiG Logan MankinsG Brian WatersDE Andre CarterDT Vince WilforkST Matthew SlaterAP All-ProsWR Wes Welk...

 

Heckler & Koch MP5

MP5 MP5A3 Jenis Pistol mitraliur Negara asal Jerman Sejarah pemakaian Masa penggunaan 1966–sekarang Sejarah produksi Perancang Tilo Möller, Manfred Guhring,Georg Seidl, Helmut Baureuter Tahun 1964 Produsen Heckler & Koch Diproduksi 1966— Spesifikasi Berat 2,54 kg (MP5A2)2,88 kg (MP5A3) Panjang Popor tetap: 680 mm (26.8 in) Popor lipat: 490 mm (masuk) 660 mm (keluar) Panjang laras 225 mm Peluru 9 x 19 mm Luger Mekanisme Semburan Kebelakang, bolt tertutup Rata² te...

Lancaster pistol

Multi-barrel pistol Lancaster pistol Break action Lancaster pistol on display at the Royal Armouries Museum in LeedsTypeMulti-barrel pistolPlace of origin United KingdomService historyWarsAnglo-Zulu War First Boer War Mahdist War Second Boer War World War IProduction historyDesignerCharles W. Lancaster and Henry ThornDesignedc. 1860Producedmid to late 19th centurySpecificationsCartridge.38 S&W.450 Adams.455 Webley.577 SniderCalibre.380 inch.450 inch.455 inch.577 inchBarrels...

 

Low Life (film)

Low LifePoster untuk Low Life (2004)Nama lainHangul하류인생 Hanja下流人生 Alih Aksara yang DisempurnakanHaryu insaengMcCune–ReischauerHaryu insaeng SutradaraIm Kwon-taekProduserLee Tae-wonDitulis olehIm Kwon-taekPemeranCho Seung-wooPenata musikShin Jung-hyeonSinematograferJung Il-sungPenyuntingPark Soon-dukTanggal rilis 21 Mei 2004 (2004-05-21) Durasi105 menitNegaraKorea SelatanBahasaKorea Low Life (Hangul: 하류인생; RR: Haryu insaeng; juga...

 

Cowlitz Indian Tribe

Indian tribe in Washington, United States Ethnic group Cowlitz Indian TribeTraditional Cowlitz territoryTotal population3500 + enrolled members[1]Regions with significant populations United States ( Washington)LanguagesEnglish, Cowlitz[2]Religiontraditional tribal religionRelated ethnic groupsother Cowlitz people[3] The Cowlitz Indian Tribe is a federally recognized tribe of Cowlitz people. They are a tribe of Southwestern Coast Salish and Sahaptan indigenous...

Kisbárapáti

Village in Southern Transdanubia, HungaryKisbárapátiVillage Coat of armsLocation of Somogy county in HungaryKisbárapátiLocation of KisbárapátiCoordinates: 46°36′21″N 17°52′00″E / 46.60584°N 17.86656°E / 46.60584; 17.86656Country HungaryRegionSouthern TransdanubiaCountySomogyDistrictTabRC DioceseKaposvárArea • Total28.71 km2 (11.08 sq mi)Population (2017) • Total348[1]DemonymkisbárapátiTime zo...

 

Willingdon and Jevington

Human settlement in EnglandWillingdon and JevingtonRed Lion, WillingdonWillingdon and JevingtonLocation within East SussexArea10.6 km2 (4.1 sq mi) [1]Population7,440 (2011)[2]• Density1,511/sq mi (583/km2)OS grid referenceTQ590024• London51 miles (82 km) NNWCivil parishWillingdon and JevingtonDistrictWealdenShire countyEast SussexRegionSouth EastCountryEnglandSovereign stateUnited KingdomPost townEASTBOURNEP...

 

Small Sale (EP)

2001 EP by MirahSmall Sale EPEP by MirahReleased2001Recorded2000/2001GenreIndie rockLengthApproximately 12 min. LabelModern Radio Records (7)ProducerBobby Burg, Mirah, Diana ArensMirah chronology Advisory Committee(2001) Small Sale EP(2001) Cold Cold Water EP(2002) Small Sale EP is a 2001 album by Mirah, released on Modern Radio Records. It was positively received by Allmusic, who stated Mirah's sweetly expressive, breathy voice...is intoxicatingly endearing, as are the electronic bea...

كأس الاتحاد الأوروبي 1981–82

يفتقر محتوى هذه المقالة إلى الاستشهاد بمصادر. فضلاً، ساهم في تطوير هذه المقالة من خلال إضافة مصادر موثوق بها. أي معلومات غير موثقة يمكن التشكيك بها وإزالتها. (أغسطس 2019) كأس الاتحاد الأوروبي 1981–82 تفاصيل الموسم الدوري الأوروبي  النسخة 11  التاريخ بداية:16 سبتمبر 1981  نها�...

 

1840 Arkansas gubernatorial election

1840 Arkansas gubernatorial election ← 1836 3 August 1840 1844 →   Nominee Archibald Yell Party Democratic Popular vote 10,554 Percentage 96.36% Governor before election James Sevier Conway Democratic Elected Governor Archibald Yell Democratic Elections in Arkansas Federal government Presidential elections 1836 1840 1844 1848 1852 1856 1860 1868 1872 1876 1880 1884 1888 1892 1896 1900 1904 1908 1912 1916 1920 1924 1928 1932 1936 1940 1944 1948 1952 1956 1960 1...