2023大數(shù)據(jù)入門指南

大數(shù)據(jù)入門指南2023目錄【1】大數(shù)伊始… 3【1,1】最簡單的運(yùn)算… 3【1,2】以0結(jié)尾的東西… 5【1,3】經(jīng)典數(shù)字… 14【1,4】增長率… 15【2】超越乘ti… 17【2,1】感嘆號(hào)! 17【2,2】沒有運(yùn)算符… 18【2,3】阿克曼函數(shù)… 20【2,4】多邊形符號(hào)… 22【3】箭頭時(shí)代… 28【3,1】高德納上箭頭… 28【3,2】葛立恒數(shù)… 29【3,3】低級(jí)超運(yùn)算… 33【3,4】康威鏈?zhǔn)郊?hào)… 35【4】超越箭頭… 39【4,1】Friedman序列… 39【4,2】Circle函數(shù) 40【4,3】Hydra函數(shù) 41【4,4】E與# 41【5】數(shù)陣時(shí)代… 65【5,1】BEAF 65【5,2】鳥之記號(hào)… 86【6】超越數(shù)陣… 101【6,1】圖論問題… 101【6,2】Loader.c 104【6,3】Σ函數(shù)… 106【6,4】Ξ函數(shù)… 108【7】序數(shù)時(shí)代… 111【7,1】f(x)與ω 111【7,2】ε、ζ和η 117【7,3】φ函數(shù)… 123【7,4】ψ函數(shù)… 130【7,5】CK 148【8】最終章… 151【8,1】Rayo數(shù)… 151【8,2】大數(shù)的意義… 152【】大數(shù)伊始【1,1】最簡單的運(yùn)算【1,1,1】正整數(shù)與后繼數(shù)正整數(shù)集N*是這么定義的,它滿足下面的所有條件:aa+有一個(gè)正整數(shù),沒有什么正整數(shù)的后繼數(shù)是它.11a,都是某一個(gè)正整數(shù)的后繼數(shù)1∈S,且(n∈S,n+∈S),S=N*問題來了.到現(xiàn)在為止,我們并不知道”1的后繼數(shù)”、”1的后繼數(shù)的后繼數(shù)”等該用什么表示.難道是”1+”、”1++”之類的嗎?于是,很久很久以前,就有人定義了:1+=2,2+=3,3+=4,4+=5,5+=6,6+=7,7+=8,8+=9,9+=10.我們用把兩個(gè)數(shù)字拼起來做成的”10”來表示9的后繼數(shù).接下來,10+=11,11+=12,…,19+=20,…,99+=100,…于是,我們就有理論上能夠表示出任何正整數(shù)的辦法了.但是,后繼數(shù)這種運(yùn)算本身只是簡單的數(shù)數(shù)而已,并無任何實(shí)際應(yīng)用價(jià)值.真正有價(jià)值的東西是下面的——【1,1,2】加法加法的定義如下,用加號(hào)”+”表示.1.a+1=a+2.a+b+=(a+b)+很簡單吧!不過,這個(gè)定義十分抽象,其實(shí)我們可以得到下面的結(jié)論a+b=a++…+(b個(gè)后繼數(shù)符號(hào)”+”).這就清晰明了多了.不過,有時(shí)候加法還是不夠用.于是,我們需要——【1,1,3】乘法乘法的定義如下,用乘號(hào)”×”表示.1.a×1=a2.a×b+=a×b+a這也是很簡單的定義.通俗一點(diǎn)說,a×b=a+a+…a(b個(gè)”a”).可以發(fā)現(xiàn),10×10=100,100×10=1000,100…0(n個(gè)”0”)×10=100…0(n+1個(gè)”0”).這將引導(dǎo)我們進(jìn)入下一級(jí)運(yùn)算——【1,1,4】乘ti乘ti運(yùn)算符一般用”^”表示,定義為:1.a^1=1a^b+=a^b×a.一般來說,它是右結(jié)合的,a^b^ca^(b^c)有時(shí)候,乘法可以不用任何運(yùn)算符來表示,直接用上標(biāo)記作”ab”,可以”通俗一點(diǎn)”說,ab=a×a×…a(b個(gè)”a”).然而,更高級(jí)的運(yùn)算很長時(shí)間沒有得到發(fā)展,一直到近代.人們曾經(jīng)”創(chuàng)造”了很多數(shù)字,給它們命名,用以說出任何可能的正整數(shù).【1,2】以”0”結(jié)尾的東西【1,2,1】4種中文計(jì)數(shù)系統(tǒng)在中文數(shù)字中,10稱為十,100稱為百,1000稱為千,10000稱為萬.這是不可置疑的.但是,更大的數(shù)字該如何稱呼呢?數(shù)字名稱下數(shù)萬進(jìn)中數(shù)上數(shù)萬10^410^410^410^4億10^510^810^810^8兆10^610^1210^1610^16京10^710^1610^2410^32垓10^810^2010^3210^64秭10^910^24數(shù)字名稱下數(shù)萬進(jìn)中數(shù)上數(shù)萬10^410^410^410^4億10^510^810^810^8兆10^610^1210^1610^16京10^710^1610^2410^32垓10^810^2010^3210^64秭10^910^2410^4010^128穰10^1010^2810^4810^256溝10^1110^3210^5610^512澗10^1210^3610^6410^1024正10^1310^4010^7210^2048載10^1410^4410^8010^4096由此可見,”下數(shù)”實(shí)際上是”十進(jìn)”,”中數(shù)”則是”萬萬進(jìn)”,而”上數(shù)”是”平ti進(jìn)”.其實(shí),”上數(shù)”是一種很經(jīng)濟(jì)的表示ti法,它可以用最少的稱呼表示出大數(shù).與“上數(shù)”類似的表示法還有很多,如下面一例.【1,2,2】華嚴(yán)大數(shù)如下表所示.也是”平ti進(jìn)”.名稱數(shù)值名稱數(shù)值倶胝107一持10469762048阿庾多1014異路10939524096那由他1028顛倒101879048192頻波羅1056三末耶103758096384矜羯羅10112毗睹羅107516192768阿伽羅10224奚婆羅1015032385536最勝10448伺察1030064771072摩婆羅10896周廣1060129542144阿婆羅101792高出10120259084288多婆羅103584最妙10240518168576界分107168泥羅婆10481036337152普摩1014336訶理婆10962072674304禰摩1028672一動(dòng)101924145348608阿婆鈐1057344訶理蒲103848290697216彌伽婆10114688訶理三107696581394432毗攞伽10229376奚魯伽1015393162788864毗伽婆10458752達(dá)攞歩陀1030786325577728僧羯邏摩10917504訶魯那1061572651155456毗薩羅101835008摩魯陀10123145302310912毗贍婆103670016懺慕陀10246290604621824毗盛伽107340032瑿攞陀10492581209243648毗素陀1014680064摩魯摩10985162418487296毗婆訶1029360128調(diào)伏101970324836974592毗薄底1058720256離憍慢103940649673949184毗佉擔(dān)10117440512不動(dòng)107881299347898368稱量10234881024極量1015762598695796736名稱數(shù)值名稱數(shù)值阿么怛羅1031525197391593472阿怛羅102115620184325601055735808勃么怛羅1063050394783186944酰魯耶104231240368651202111471616伽么怛羅10126100789566373888薜魯婆108462480737302404222943232那么怛羅10252201579132747776羯羅波1016924961474604808445886464奚么怛羅10504403158265495552訶婆婆1033849922949209616891772928鞞么怛羅101008806316530991104毗婆羅1067699845898419233783545856鉢羅么怛羅102017612633061982208那婆羅10135399691796838467567091712尸婆么怛羅104035225266123964416摩攞羅10270799383593676935134183424翳羅108070450532247928832娑婆羅10541598767187353870268366848薜羅1016140901064495857664迷攞普101083197534374707740536733696諦羅1032281802128991715328者么羅102166395068749415481073467392偈羅1064563604257983430656駄么羅104332790137498830962146934784窣步羅10129127208515966861312鉢攞么陀108665580274997661924293869568泥羅10258254417031933722624毗迦摩1017331160549995323848587739136計(jì)羅10516508834063867445248烏波跋多1034662321099990647697175478272細(xì)羅101033017668127734890496演説1069324642199981295394350956544睥羅102066035336255469780992無盡10138649284399962590788701913088謎羅104132070672510939561984出生10277298568799925181577403826176娑攞荼108264141345021879123968無我10554597137599850363154807652352謎魯陀1016528282690043758247936阿畔多101109194275199700726309615304704契魯陀1033056565380087516495872青蓮華10^(2^98×7)摩睹羅1066113130760175032991744鉢頭摩10^(2^99×7)娑母羅10132226261520350065983488僧祇10^(2^100×7)阿野娑10264452523040700131966976趣10^(2^101×7)迦么羅10528905046081400263933952至10^(2^102×7)摩伽婆101057810092162800527867904阿僧祇10^(2^103×7)名稱數(shù)值阿僧祇轉(zhuǎn)10141976867225561692967630759002112無量10283953734451123385935261518004224無量轉(zhuǎn)10567907468902246771870523036008448無邊101135814937804493543741046072016896無邊轉(zhuǎn)102271629875608987087482092144033792無等104543259751217974174964184288067584無等轉(zhuǎn)109086519502435948349928368576135168不可數(shù)1018173039004871896699856737152270336不可數(shù)轉(zhuǎn)1036346078009743793399713474304540672不可稱1072692156019487586799426948609081344不可稱轉(zhuǎn)10145384312038975173598853897218162688不可思10290768624077950347197707794436325376不可思轉(zhuǎn)10581537248155900694395415588872650752不可量101163074496311801388790831177745301504不可量轉(zhuǎn)102326148992623602777581662355490603008不可說104652297985247205555163324710981206016不可說轉(zhuǎn)109304595970494411110326649421962412032不可說不可說1018609191940988822220653298843924824064不可說不可說轉(zhuǎn)10^(2^122×7)=1037218383881977644441306597687849648128什么?你認(rèn)為這種數(shù)字很大?那么你的目光真是夠短淺的……【1,2,3】2種”-illion”數(shù)字系統(tǒng)我們知道,英文中的ten是10,hundred是100,thousand是1000,而million106.2種計(jì)數(shù)系統(tǒng)就從這里開始分離.它們是美式英文的”shortscale”和其他外文的”longscale”.shortscale103n+3命名以-illion結(jié)尾的數(shù)字,名稱shortscalelongscale名稱longscalemillion10^610^6milliard10^9billion10^910^12billiard10^15trillion10^1210^18trilliard10^21quadrillion10^1510^24quadrilliard10^27quintillion10^18名稱shortscalelongscale名稱longscalemillion10^610^6milliard10^9billion10^910^12billiard10^15trillion10^1210^18trilliard10^21quadrillion10^1510^24quadrilliard10^27quintillion10^1810^30quintilliard10^33sextillion10^2110^36sextilliard10^39septillion10^2410^42septilliard10^45octillion10^2710^48octilliard10^51nonillion10^3010^54nonilliard10^57decillion10^3310^60decilliard10^63undecillion10^3610^66undecilliard10^69duodecillion10^3910^72duodecilliard10^75tredecillion10^4210^78tredecilliard10^81quattuordecillion10^4510^84quattuordecilliard10^87quindecillion10^4810^90quindecilliard10^93sexdecillion10^5110^96sexdecilliard10^99septendecillion10^5410^102septendecilliard10^105octodecillion10^5710^108octodecilliard10^111novemdecillion10^6010^114novemdecilliard10^117vigintillion10^6310^120vigintilliard10^123unvigintillion10^6610^126unvigintilliard10^129duovigintillion10^6910^132duovigintilliard10^135當(dāng)然,這些數(shù)字并不止這么點(diǎn).我們還有名稱shortscalelongscaletrigintillion10^9310^180quadragintillion10^12310^240quinquagintillion10^15310^300sexagintillion10^18310^360名稱shortscalelongscaleseptuagintillion10^21310^420octogintillion10^24310^480nonagintillion10^27310^540centillion10^30310^600primo-vigesimo-centillion10^36610^726ducentillion10^60310^1200trecentillion10^90310^1800quadringentillion10^120310^2400quingentillion10^150310^3000sescentillion10^180310^3600septingentillion10^210310^4200octingentillion10^240310^4800nongentillion10^270310^5400millillion10^300310^6000dumillillion10^600310^12000myrillion10^3000310^60000milli-millillion(micrillion)10^300000310^6000000nanillion10^300000000310^6000000000picillion10^300000000000310^6000000000000femtillion103000000000000003106000000000000000attillion103000000000000000003106000000000000000000zeptillion103000000000000000000003106000000000000000000000yoctillion10^(3×10^24+3)10^(6×10^24)xonillion10^(3×10^27+3)10^(6×10^27)vecillion10^(3×10^30+3)10^(6×10^30)mecillion10^(3×10^33+3)10^(6×10^33)duecillion10^(3×10^36+3)10^(6×10^36)trecillion10^(3×10^39+3)10^(6×10^39)看,trecillion已經(jīng)超越華嚴(yán)經(jīng)數(shù)字了,然而我們還有更大的.以下均為shortscale中的數(shù)字.名稱shortscale名稱shortscaleTetrecillion10^(3×10^42+3)Exdakillion103×10^(3×10^48)+3Pentecillion10^(3×10^45+3)Zedakillion103×10^(3×10^51)+3Hexecillion10^(3×10^48+3)Yodakillion103×10^(3×10^54)+3Heptecillion10^(3×10^51+3)Nedakillion103×10^(3×10^57)+3Octecillion10^(3×10^54+3)Ikillion103×10^(3×10^60)+3Ennecillion10^(3×10^57+3)Ikenillion103×10^(3×10^63)+3Icosillion10^(3×10^60+3)Icodillion103×10^(3×10^66)+3Triacontillion10^(3×10^90+3)Ictrillion103×10^(3×10^69)+3Tetracontillion10^(3×10^120+3)Icterillion103×10^(3×10^72)+3Pentacontillion10^(3×10^150+3)Icpetillion103×10^(3×10^75)+3Hexacontillion10^(3×10^180+3)Ikectillion103×10^(3×10^78)+3Heptacontillion10^(3×10^210+3)Iczetillion103×10^(3×10^81)+3Octacontillion10^(3×10^240+3)Ikyotillion103×10^(3×10^84)+3Ennacontillion10^(3×10^270+3)Icxenillion103×10^(3×10^87)+3Hectillion10^(3×10^300+3)Trakillion103×10^(3×10^90)+3Killillion10^(3×10^3000+3)Tekillion103×10^(3×10^120)+3Megillion103×10^3000000+3Pekillion103×10^(3×10^150)+3Gigillion103×10^3000000000+3Exakillion103×10^(3×10^180)+3Terillion103×10^3000000000000+3Zakillion103×10^(3×10^210)+3Petillion103×10^(3×10^15)+3Yokillion103×10^(3×10^240)+3Exillion103×10^(3×10^18)+3Nekillion103×10^(3×10^270)+3Zettillion103×10^(3×10^21)+3Hotillion103×10^(3×10^300)+3Yottillion103×10^(3×10^24)+3Botillion103×10^(3×10^600)+3Xennillion103×10^(3×10^27)+3Trotillion103×10^(3×10^900)+3Dakillion103×10^(3×10^30)+3Totillion103×10^(3×10^1200)+3Hendillion103×10^(3×10^33)+3Potillion103×10^(3×10^1500)+3Dokillion103×10^(3×10^36)+3Exotillion103×10^(3×10^1800)+3Tradakillion103×10^(3×10^39)+3Zotillion103×10^(3×10^2100)+3Tedakillion103×10^(3×10^42)+3Yootillion103×10^(3×10^2400)+3Pedakillion103×10^(3×10^45)+3Notillion103×10^(3×10^2700)+3名稱shortscale名稱shortscaleKalillion103×10^(3×10^3000)+3Hotejillion103×10^(3×10^300000000)+3Dalillion103×10^(3×10^6000)+3Gijillion103×10^(3×10^3000000000)+3Tralillion103×10^(3×10^9000)+3Astillion103×10^(3×10^(3×10^12))+3Talillion103×10^(3×10^12000)+3Lunillion103×10^(3×10^(3×10^15))+3Palillion103×10^(3×10^15000)+3Fermillion103×10^(3×10^(3×10^18))+3Exalillion103×10^(3×10^18000)+3Jovillion103×10^(3×10^(3×10^21))+3Zalillion103×10^(3×10^21000)+3Solillion103×10^(3×10^(3×10^24))+3Yalillion103×10^(3×10^24000)+3Betillion103×10^(3×10^(3×10^27))+3Nalillion103×10^(3×10^27000)+3Glocillion103×10^(3×10^(3×10^30))+3Dakalillion103×10^(3×10^30000)+3Gaxillion103×10^(3×10^(3×10^33))+3Hotalillion103×10^(3×10^300000)+3Supillion103×10^(3×10^(3×10^36))+3Mejillion103×10^(3×10^3000000)+3Versillion103×10^(3×10^(3×10^39))+3Dakejillion103×10^(3×10^30000000)+3Multillion103×10^(3×10^(3×10^42))+3Bentrizillion=106×10^(6×10^(6×10^6000000000))【1,2,4】”-yllion”記法名稱數(shù)值名稱數(shù)值myriad10^4septyllion10^512myllion10^8octyllion10^1024byllion10^16nonyllion10^2048tryllion10^32decyllion10^4096quadryllion10^64undecyllion10^8192quintyllion10^128duodecyllion10^16384sextyllion10^256tredecyllion10^32768高德納(DonaldKnuth)名稱數(shù)值名稱數(shù)值myriad10^4septyllion10^512myllion10^8octyllion10^1024byllion10^16nonyllion10^2048tryllion10^32decyllion10^4096quadryllion10^64undecyllion10^8192quintyllion10^128duodecyllion10^16384sextyllion10^256tredecyllion10^32768名稱數(shù)值名稱數(shù)值quattuordecyllion10^65536novemtrigintyllion10^2^41quindecyllion10^131072quadragintyllion10^2^42sexdecyllion10^262144quinquagintyllion10^2^52septdecyllion10^524288sexagintyllion10^2^62octodecyllion101048576septuagintyllion10^2^72novemdecyllion102097152octogintyllion10^2^82vigintyllion104194304nonagintyllion10^2^92unvigintyllion108388608centyllion10^2^102duovigintyllion1016777216ducentyllion10^2^202trevigintyllion1033554432trecentyllion10^2^302quattuorvigintyllion1067108864quadringentyllion10^2^402quinvigintyllion10134217768quingentyllion10^2^502sexvigintyllion10268435456sescentyllion10^2^602septemvigintyllion10534870912septingentyllion10^2^702octovigintyllion101073741824octingentyllion10^2^802novemvigintyllion102147483648nongentyllion10^2^902trigintyllion104294967296millyllion10^2^1002untrigintyllion108589934592myryllion10^2^10002duotrigintyllion1017179869184micryllion10^2^1000002tretrigintyllion1034359738368nanyllion102^1000000002quattuortrigintyllion1068719476736picyllion102^1000000000002quintrigintyllion10137438953472femtyllion102^(10^15+2)sextrigintyllion10274877906944attyllion102^(10^18+2)septtrigintyllion10549755813888zeptyllion102^(10^21+2)octotrigintyllion101099511627776yoctyllion102^(10^24+2)應(yīng)當(dāng)指出的是,如果Bentrizyllion存在的話,那得多大!【1,3】經(jīng)典數(shù)字【1,3,1】e

163e 163被稱作拉馬努金數(shù),2625374126407687442.5×10-13,以致于人們會(huì)誤認(rèn)為它是個(gè)整數(shù).【1,3,2】魔ti8!×37×12!×211/2=43252003274489856000,這是三階魔ti的排列ti法總數(shù)7!×36×24!×24!/246≈7.4011968415649×1045,這是四階魔ti的排列ti法總數(shù).五階魔ti為8!×37×12!×210×24!×(24!/246)2≈2.8287094227774×1074,六階魔ti7!×36×24!2×(24!/246)4≈1.5715285840103×10116,七階魔ti8!×37×12!×210×24!2×(24!/246)6≈1.9500551183732×10160.【1,3,3】數(shù)獨(dú)220×38×5×7×27704267971=6670903752021072936960,這是9×9數(shù)獨(dú)的總種數(shù).【1,3,4】阿伏伽德羅常數(shù)阿伏伽德羅常數(shù)約為6.02214179×1023【1,3,5】大數(shù)定律2.26881×1039,一個(gè)神秘的大數(shù).【1,3,6】帶電粒子136×2257≈3.149544×1079個(gè).【1,3,7】斯奎斯數(shù)π(x)x的質(zhì)數(shù)的個(gè)數(shù),π(2)=1π(3)=2π(10)=4等.我們可以用一個(gè)積分來近似估計(jì)它li()=xdt

.我們發(fā)現(xiàn),對于”比較小”0lnt的數(shù)x,總是有π(x)<li(x),那么,對于任意的x,還是否會(huì)有π(x)<li(x)成立呢?斯奎斯證明了π(x)li(x)的函數(shù)值會(huì)”交叉”無數(shù)次!而且,正確的,那么第一個(gè)”交叉點(diǎn)”e^e^e^79≈10^(3.5536897484442191×108852142197543270606106100452735038),這個(gè)數(shù)稱作第一斯奎斯數(shù);如果不涉及黎曼假設(shè),那么第一個(gè)”交叉點(diǎn)”e^e^e^e^7.705≈10^10^(3.29994322×10963),這個(gè)數(shù)稱作第二斯奎斯數(shù).后來,上面的結(jié)果被大大地改進(jìn)了.人們得出,第一斯奎斯數(shù)應(yīng)不超過e^e^1236.最后,第一個(gè)”交叉點(diǎn)”實(shí)際上約為1.397162914×10316.(到這里,你應(yīng)該知道了,這斯奎斯數(shù)其實(shí)是虛大的)后來S.Knapowski對這個(gè)問題進(jìn)一步研究,得出:e^e^e^e^e^35x,x的”交叉點(diǎn)”ln(ln(ln(lnx)))/e35.【1,4】增長率f的增長率,f(x)的大小,xf(x)的相對大小.f(x)=10000xg(x)=x2相比,0<x<10000f(x)>g(x),f(x)g(x).實(shí)際上,xf(x)<g(x),f(x)g(x).這有點(diǎn)像高、低階無窮大.但是,增長率與高、低階無窮大又有一點(diǎn)不同.f(x)=3xg(x)=2x都是指數(shù)函數(shù),f(x)g(x)高階的無窮大,但是它們還是有可比性的.從大類來看,lnlnlnlnlnln(x)的增長率就很小,lnlnlnlnln(x),…,ln(x),線性函數(shù),接著xlnln(x)xln(x)那樣的函數(shù),x1.000001.順著冪函數(shù),x1.0001,x1.001,x1.1,x1.2,x2,x3,x10000之類的函數(shù).1.001x,1.1x,2x,1000x之類的指數(shù)函數(shù),xx,2x^1.01,2x^1.1,2x^2,2x^100之類的函數(shù),2x^x,xx^x,22^x^2,x^x^x^x,x^x^x^x^x,等等.f(x)=x^xx^x^x^xg(x)=x^x2(x^x^x^x)更低階的無窮大,但是從"大類"來看,它們還是有"相近的"增長率.這里和后面提到函數(shù)的增長率,在第【7】章會(huì)有一個(gè)統(tǒng)一的比較.【】超越乘ti【2,1】感嘆號(hào)!【2,1,1】普通運(yùn)算感嘆號(hào)”!”表示一種運(yùn)算,稱作階乘.其定義如下:1.0!=12.a!=(a-1)!×a我們很容易知道,a!=1×2×3×…a.這就是”階乘”這個(gè)名字的意義.實(shí)際上,階乘可以擴(kuò)展到全實(shí)數(shù)范圍(負(fù)整數(shù)除外).【2,1,2】Γ函數(shù)對于非負(fù)整數(shù)n,你可以認(rèn)為n!=Γ(n+1).這個(gè)Γ函數(shù)還可以如下定義.Γ(a)

=∞ta1e-tdt0它有這樣一個(gè)遞推公式,就是Γ(n+1)=nΓ(n).利用它,我們可以計(jì)算負(fù)數(shù)a的0Γ函數(shù)值.階乘的增長與乘ti相近,可以算出,100!≈9.3326215443×10157,而21000!≈4.0238726007×102567.還有(?)!=π,(-?)!= π等.2【2,1,3】雙階乘你是否認(rèn)為,3的數(shù)迭代多次階乘,會(huì)變得很大呢?沒錯(cuò).不過你是否認(rèn)為,10!!>10!呢?這回你就錯(cuò)了.實(shí)際上,10!!中的”!!”是一個(gè)運(yùn)算符,它表示雙階乘,就是”隔著階乘”.實(shí)際上,10!!=2×4×6×8×10=3840,10!=3628800,所10!!<10!.一般的,n!!=1×3×5×…n,n!!=2×4×6×…n.【2,1,4】迭代階乘n!2來表示大家所想象的(n!)!,n!3((n!)!)!,一般地,n!1=n!,而n!m+1=(n!m)!.這就是迭代階乘.注意,n!mn!m次ti,而(3!)^2=36.【2,2】沒有運(yùn)算符【2,2,1】”最大的數(shù)”游戲設(shè)想一下這個(gè)游戲.用若干個(gè)給定的數(shù)字,不用任何運(yùn)算符,你能組成最大的數(shù)是什么?給你4個(gè)”1”,你可以組出像1111,1111,1111,1111,1111之類的數(shù).不過后三者是毫無意義的,1次ti都是它自己,1的任何數(shù)次ti1.費(fèi)很多個(gè)”1”.實(shí)際上,1111=285311670611.4個(gè)”2”,2222是最大的數(shù)了.

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<222因此,2222才是最大的一個(gè)數(shù).順便說一下,2222≈2.0650635398359×101262611.如果是4個(gè)”3”呢?這時(shí)3333仍然是最大的.3333≈4.685801298241×102652345952577568,這個(gè)數(shù)介于調(diào)伏和離憍慢之間.不過,如果是4個(gè)”4”,那么因?yàn)?4>44,所以最大的數(shù)應(yīng)該是4444而不是4444這里,4444≈2.36102267146×10^8072304726028225379382630397085399030071367921738743031867082828418414481568309149198911814701229483451981557574771156496457238535299087481244990261351116,pentacontillionhexacontillion之間.現(xiàn)在,考慮一下給出5個(gè)”2”的情形.我們可以組出的一個(gè)數(shù)是22222,它大約106.216460684426429×10^1262610,killillion和megillion之間.大的嗎?【2,2,2】第四級(jí)運(yùn)算很可惜,即使不用任何”標(biāo)點(diǎn)符號(hào)”,我們也有比這更大的數(shù).下面是一個(gè)不用任何運(yùn)算符的超越了加法、乘法和乘ti的運(yùn)算的定義:11aa2a1ab如果說加法是第一級(jí)運(yùn)算,乘法是第二級(jí)運(yùn)算,乘ti是第三級(jí)運(yùn)算,那么這種a用”ba”來示的運(yùn)算,就稱作第四級(jí)運(yùn)算.通俗一點(diǎn)說,ba=aaN (共b個(gè)”a”).于是剛才的問題就可以進(jìn)一步擴(kuò)展了.4個(gè)”1”1111,1111^11^11^11^11^11^11^11^11^11,Bentrizillion,你大概沒見過這種數(shù)字吧.4個(gè)”2”可以組成,這個(gè)數(shù)字應(yīng)該這么表達(dá):2^2^…2(2^2^…2(22個(gè)”2”)個(gè)”2”).很吃驚嗎4個(gè)

3可以組成 3,可以說成3^3^…3(共3^3^…3(共7625597484987個(gè)”3”)個(gè)”3”),這里33=3^3^3=3^27>34=81>33.最后,5個(gè)”2”可以組成22222,2^2^…2(共個(gè)”2”),你得自己想象好一陣子.【2,2,3】實(shí)數(shù)擴(kuò)展設(shè)想一下,yx=x^a^b^c^…1^1^1^…,abc等數(shù)依次減小,到后面幾乎1.那么,這樣的”減小”應(yīng)該如何進(jìn)行呢?1+erf(2z-1)

x﹣t2首先,讓我們定義一個(gè)函數(shù):g(x,z)=x

,其中erf(x)

=0

dt.于是,我們可g(x,z)1x的”平滑”過渡.下面是一些特殊值(近似值).g(x,z)z=-1z=0z=1/2z=1z=2x=21.000007661.056029111.414213561.893887191.99998469x=31.000012131.090248251.732050812.751666882.99996360x=101.000025431.198531923.162277668.343540859.99974568n→∞然后,T(x,y,n)=g(x,y)^g(x,y-1)^…g(x,y-n+2)^g(x,y-n+1),再取極限h(x,y)=limT(x,y,n),h(x,y)似乎可以作為第四級(jí)運(yùn)算的近似.不過,如果真正計(jì)算起來,n→∞等不準(zhǔn)確的結(jié)果,h(2,4)≈30815.4026169,65536相差一倍多,而h(3,4)≈2.43×105924531213185,1.25638×103638334640024相2286196573161個(gè)數(shù)量級(jí)!實(shí)際上,第四級(jí)運(yùn)算的實(shí)數(shù)擴(kuò)展,很多人嘗試過,但都沒有加法、乘法、乘ti的擴(kuò)展那么”平滑”(即可導(dǎo)性).即便”平滑”,也像剛才一例一樣,失去了原有的整數(shù)運(yùn)算結(jié)果.這個(gè)”擴(kuò)展”之路還很長很長.【2,3】阿克曼函數(shù)【2,3,1】二元函數(shù)二元阿克曼函數(shù)一般用Ack(m,n)表示,其定義如下:1.Ack(0,n)=n+12.Ack(m,0)=Ack(m-1,1)Ack(m,n)=Ack(m-1,Ack(m,n-1))這樣,可以認(rèn)為,Ack(1,0)=Ack(0,1)=2,Ack(1,x)=Ack(0,Ack(1,x-1))=Ack(1x-1)+1,Ack(1,x)=Ack(1,0)+x=x+2.接著,Ack(2,x)=Ack(1,Ack(2,x-1))=Ack(2,x-1)+2=…=Ack(2,0)+2x=Ack(1,1)+2x=2x+3.到這里,似乎還沒有大數(shù)或者增長很快的函數(shù)出現(xiàn).不過,如果往下計(jì)算,么這個(gè)增長將會(huì)很快.Ack(3,x)=Ack(2,Ack(3,x-1))=2×Ack(3,x-1)+3,可知Ack(3,x)+3=2×(Ack(3,x-1)+3).Ack(3,0)=Ack(2,1)=5,Ack(3,x)=2x×8-3=2x+3-3.Ack(4,x)+3=Ack(3,Ack(4,x-1))+3=2Ack(4,x-1)+3,Ack(4,0)=Ack(3,1)=13,因Ack(4,x)+3=2^2^…2^(13+3)(x個(gè)”2”)=2^2^…2^(2^2^2)=x+32,Ack(4,x)=x+32-3.到這里,你應(yīng)該已經(jīng)能感受到這個(gè)函數(shù)的增長之快了.我們將一些有特殊值列舉如下:Ack(m,n)01234m=501235136553312351365533655362-3234729265536-365536334596162-3655362223﹣4561112572-3655362225671325382-3655362223nn+1n+22n+32n+3-3n+32-3【2,3,2】一元函數(shù)Friedman對此函數(shù)進(jìn)行了擴(kuò)展.一般說來,一元函數(shù)總是比二元函數(shù)更容易處理一些.Ack(n)=Ack(n,n).Ack(n)具有反函數(shù),α(n).實(shí)際上,Ack(0)=1,Ack(1)=3,Ack(2)=7,Ack(3)=61,Ack(4)=72-3,因此α(1)=0,α(3=1,α(7)=2,α(61)=3,α(72-3)=4.可見這個(gè)α函數(shù)的增長是非常慢的.【2,4】多邊形符號(hào)【2,4,1】、□與○斯坦豪斯提出了用、□和表示的符號(hào).他定義:n放進(jìn)一個(gè)△nnn,把nn放進(jìn)n個(gè)△中,n放進(jìn)一個(gè)○nnn個(gè)□中.也就是說n=nnn

,n= n

(n層三角形、n層正ti形).例如,MEGA

,MEGA=

2 = =

,所以,它2=2=42256256層三角形里面,y=xx256次.這個(gè)數(shù)2=2=42幾乎與第四級(jí)運(yùn)算的數(shù)有相同的”檔次”.3=3=327GrandMega等于 = ,這個(gè)數(shù)比Mega大得多了!3=3=3274Great4

,GongMega=

,

,Heptomega=7,56Octomega8Nonomega9megistron10不過,Ack(5,14)相比56真是小得可憐,Ack(5,14)Ack(6,1)=Ack(7,0).2Megision,也叫A-ooga,等于 ,也就是把Mega放進(jìn)一個(gè)圓里面.接著,2GrandMegision=3,GreatMegision=4,GongMegision=5,Hexomegision=6,Heptomegision=7,Octomegision=8,Nonomegision=9,610Megisiplextron= ,Ack(6,2).10megisiduon等于”23個(gè)圓里面”,Grandmegisiduon等于”3放3個(gè)圓里面”,Greatmegisiduon等于”43個(gè)圓里面”,Gongmegisiduon等于”53個(gè)圓里面”,hexomegisiduon等于”63個(gè)圓里面”,heptomegisiduon等于”73個(gè)圓里面”,octomegisiduon等于”83個(gè)圓里面”,nonomegisiduon等于”93個(gè)圓里面”,megisiduplextron等于”103個(gè)圓里面”.megisitruon等于”24”,Grandmegisitruon等于”3放4個(gè)圓里面Greatmegisitruon等于”44個(gè)圓里面”,Gongmegisitruon等于”54個(gè)圓里面”,hexomegisitruon等于”64個(gè)圓里面”,heptomegisitruon等于”74個(gè)圓里面”,octomegisitruon等于”84個(gè)圓里面”,nonomegisitruon等于”94個(gè)圓里面”,megisitriplextron等于”104個(gè)圓里面”.megisiquadruon等于”把2放進(jìn)5個(gè)圓里面”,Grandmegisiquadruon等于”35個(gè)圓里面Greatmegisiquadruon等于”45個(gè)圓里面”,Gongmegisiquadruon等于”55個(gè)圓里面”,hexomegisiquadruon等于”65個(gè)圓里面”,heptomegisiquadruon等于”75個(gè)圓里面”,octomegisiquadruon等于”85”,nonomegisiquadruon等于”把9放進(jìn)5個(gè)圓里面megisiquadruplextron等于”把10放進(jìn)5個(gè)圓里面”.【2,4,2】當(dāng)變成五邊形莫澤(Moser)把這個(gè)記號(hào)一般化,他定義:n放進(jìn)一個(gè)△里面等于nn,nmnnm-1邊形里面.這樣,取代了原來的圓.(這個(gè)記號(hào)也就能夠表示更大的數(shù)了)A-ooga等于”2”.A-oogra等于”3Grandmegisiduon.A-oogrea等于”4放進(jìn)一個(gè)六邊形里面”Greatmegisitruon.A-oogonga等于”5放進(jìn)一個(gè)六邊形里面”,也等GongMegisiquadruon.A-oohexa等于”6放進(jìn)一個(gè)六邊形里面”,A-oohepta等于”7放進(jìn)一個(gè)六邊形里面”,A-oocta等于”8放進(jìn)一個(gè)六邊形里面”,A-ooennea等于”9放進(jìn)一個(gè)六邊形里面”,A-oomega等于”10放進(jìn)一個(gè)六邊形里面”.A-oogatiplex,Betomega,等于”22個(gè)六邊形里面”,也就是”把2放進(jìn)一個(gè)七邊形里面”.A-oogatiduplex等于”23個(gè)六邊形里面”,A-oogatitriplex等于”24個(gè)六邊形里面”,A-oogatiquadruplex等于”25個(gè)六邊形里面”.Betogiga,A-oogratiduplex,等于”33個(gè)六邊形里面”,或者”把3放進(jìn)一個(gè)七邊形里面”.Brantomega,flexinega,等于”22個(gè)七邊形里面”.Brantogiga等于”32個(gè)七邊形里面”,Breatomega等于”23個(gè)七邊形里面”.Breatogiga,flexitria,等于”33個(gè)七邊形里面”.Bigiatomega等于”24個(gè)七邊形里面”,Bigiatogiga等于”34個(gè)七邊形里面”,Biquadriatomega等于”25個(gè)七邊形里面”,Biquadriatogiga等于”把3放進(jìn)5個(gè)七邊形里面”,Biquintiatomega等于”把2放進(jìn)6個(gè)七邊形里面”,Biquintiatomega等于”36個(gè)七邊形里面”.最后,莫澤數(shù)(Moser數(shù))等于”2MEGA邊形里面”,這個(gè)數(shù)還是小于Ack(Ack(5)).【2,4,3】超越多邊形Aarex對這樣的記號(hào)進(jìn)行了一次擴(kuò)展,這個(gè)記號(hào)就脫離了多邊形的形狀而存在.一般用M(a,b)表示”a放進(jìn)一個(gè)b+2邊形里面”.M函數(shù),如:M(2,0,1),M(2,3,5),M(7,6,5,4,3,2,1).它們的定義如下:1.M(a,1)=aa2.M(#0)=M(#),這里用”#”表示一段數(shù)字序列,下同3.M(a,0,0,…0,0,b)(c個(gè)”0”)=M(a,0,0,…0,a,b-1)(c-1個(gè)”0”)4.M(a,b#)=M(M(…M(M(a,b-1#),b-1#)…,b-1#),b-1#)(a個(gè)”M”)比tiM(3,5,2)=M(M(M(3,4,2),4,2),4,2),M(4,0,0,1)=M(4,0,4,0)=M(4,0,4)=M(4,4,3)=M(M(M(M(4,3,3),3,3),3,3),3,3).接下來我們用這種函數(shù)把前面的數(shù)字表示出來.名稱數(shù)值名稱數(shù)值MegaM(2,3)GrandMegaM(3,3)GreatMegaM(4,3)GongMegaM(5,3)HexomegaM(6,3)HeptomegaM(7,3)OctomegaM(8,3)NonomegaM(9,3)MegistronM(10,3)megisionM(M(2,3),3)GrandmegisionM(M(3,3),3)GreatmegisionM(M(4,3),3)GongmegisionM(M(5,3),3)HexomegisionM(M(6,3),3)HeptomegisionM(M(7,3),3)OctomegisionM(M(8,3),3)NonomegisionM(M(9,3),3)MegisiplextronM(M(10,3),3)megisiduon=M(megision,3)=M(M(M(2,3),3),3)Grandmegisiduon=M(Grandmegision,3)=M(M(M(3,3),3),3)Greatmegisiduon=M(Greatmegision,3)=M(M(M(4,3),3),3)Gongmegisiduon=M(Gongmegision,3)=M(M(M(5,3),3),3)Hexomegisiduon=M(Hexomegision,3)=M(M(M(6,3),3),3)Heptomegisiduon=M(Heptomegision,3)=M(M(M(7,3),3),3)Octomegisiduon=M(Octomegision,3)=M(M(M(8,3),3),3)Nonomegisiduon=M(Nonomegision,3)=M(M(M(9,3),3),3)Megisiduplextron=M(Megisiplextron,3)=M(M(M(10,3),3),3)megisitruon=M(megisiduon,3)=M(M(M(M(2,3),3),3),3)Grandmegisitruon=M(Grandmegisiduon,3)=M(M(M(M(3,3),3),3),3)Greatmegisitruon=M(Greatmegisiduon,3)=M(M(M(M(4,3),3),3),3)Gongmegisitruon=M(Gongmegisiduon,3)=M(M(M(M(5,3),3),3),3)Hexomegisitruon=M(Hexomegisiduon,3)=M(M(M(M(6,3),3),3),3)Heptomegisitruon=M(Heptomegisiduon,3)=M(M(M(M(7,3),3),3),3)Octomegisitruon=M(Octomegisiduon,3)=M(M(M(M(8,3),3),3),3)Nonomegisitruon=M(Nonomegisiduon,3)=M(M(M(M(9,3),3),3),3))Megisitriplextron=M(Megisiplextron,3)=M(M(M(M(10,3),3),3),3)megisiquadruon=M(M(M(M(M(2,3),3),3),3),3)Grandmegisiquadruon=M(M(M(M(M(3,3),3),3),3),3)Greatmegisiquadruon=M(M(M(M(M(4,3),3),3),3),3)Gongmegisiquadruon=M(M(M(M(M(5,3),3),3),3),3)Hexomegisiquadruon=M(M(M(M(M(6,3),3),3),3),3)Heptomegisiquadruon=M(M(M(M(M(7,3),3),3),3),3)Octomegisiquadruon=M(M(M(M(M(8,3),3),3),3),3)Nonomegisiquadruon=M(M(M(M(M(9,3),3),3),3),3)Megisiquadruplextron=M(M(M(M(M(10,3),3),3),3),3)名稱數(shù)值名稱數(shù)值A(chǔ)-oogaM(2,4)A-oograM(3,4)A-oogreaM(4,4)A-oogongaM(5,4)A-oohexaM(6,4)A-ooheptaM(7,4)A-ooctaM(8,4)A-ooenneaM(9,4)A-oomegaM(10,4)A-oogatiplexM(M(2,4),4)A-oogratiplexM(M(3,4),4)A-oogreatiplexM(M(4,4),4)A-oogongatiplexM(M(5,4),4)A-oohexatiplexM(M(6,4),4)A-ooheptatiplexM(M(7,4),4)A-ooctatiplexM(M(8,4),4)A-ooenneatiplexM(M(9,4),4)A-oomegatiplexM(M(10,4),4)A-oogatiduplex=M(A-oogatiplex,4)=M(M(M(2,4),4),4)A-oogratiduplex=M(A-oogratiplex,4)=M(M(M(3,4),4),4)A-oogreatiduplex=M(A-oogreatiplex,4)=M(M(M(4,4),4),4)A-oogongatiduplex=M(A-oogongatiplex,4)=M(M(M(5,4),4),4)A-oohexatiduplex=M(A-oohexatiplex,4)=M(M(M(6,4),4),4)A-ooheptatiduplex=M(A-ooheptatiplex,4)=M(M(M(7,4),4),4)A-ooctatiduplex=M(A-ooctatiplex,4)=M(M(M(8,4),4),4)A-ooenneatiduplex=M(A-ooenneatiplex,4)=M(M(M(9,4),4),4)A-oomegatiduplex=M(A-oomegatiplex,4)=M(M(M(10,4),4),4)A-oogatitriplex=M(A-oogatiduplex,4)=M(M(M(M(2,4),4),4),4)A-oogratitriplex=M(A-oogratiduplex,4)=M(M(M(M(3,4),4),4),4)A-oogreatitriplex=M(A-oogreatiduplex,4)=M(M(M(M(4,4),4),4),4)A-oogongatitriplex=M(A-oogongatiduplex,4)=M(M(M(M(5,4),4),4),4)A-oohexatitriplex=M(A-oohexatiduplex,4)=M(M(M(M(6,4),4),4),4)A-ooheptatitriplex=M(A-ooheptatiduplex,4)=M(M(M(M(7,4),4),4),4)A-ooctatitriplex=M(A-ooctatiduplex,4)=M(M(M(M(8,4),4),4),4)A-ooenneatitriplex=M(A-ooenneatiduplex,4)=M(M(M(M(9,4),4),4),4)A-oomegatitriplex=M(A-oomegatiduplex,4)=M(M(M(M(10,4),4),4),4)A-oogatiquadruplex=M(M(M(M(M(2,4),4),4),4),4)A-oogratiquadruplex=M(M(M(M(M(3,4),4),4),4),4)A-oogreatiquadruplex=M(M(M(M(M(4,4),4),4),4),4)A-oogongatiquadruplex=M(M(M(M(M(5,4),4),4),4),4)A-oohexatiquadruplex=M(M(M(M(M(6,4),4),4),4),4)A-ooheptatiquadruplex=M(M(M(M(M(7,4),4),4),4),4)A-ooctatiquadruplex=M(M(M(M(M(8,4),4),4),4),4)A-ooenneatiquadruplex=M(M(M(M(M(9,4),4),4),4),4)A-oomegatiquadruplex=M(M(M(M(M(10,4),4),4),4),4)名稱數(shù)值名稱數(shù)值名稱數(shù)值BetomegaM(2,5)BetogigaM(3,5)BetoteraM(4,5)BetopetaM(5,5)BetoexaM(6,5)BetozettaM(7,5)BetoyottaM(8,5)BetoxotaM(9,5)BetodakaM(10,5)BrantomegaM(M(2,5),5)BrantogigaM(M(3,5),5)BrantoteraM(M(4,5),5)BrantopetaM(M(5,5),5)BrantoexaM(M(6,5),5)BrantozettaM(M(7,5),5)BrantoyottaM(M(8,5),5)BrantoxotaM(M(9,5),5)BrantodakaM(M(10,5),5)BreatomegaM(M(M(2,5),5),5)BreatogigaM(M(M(3,5),5),5)BreatoteraM(M(M(4,5),5),5)BreatopetaM(M(M(5,5),5),5)BreatoexaM(M(M(6,5),5),5)BreatozettaM(M(M(7,5),5),5)BreatoyottaM(M(M(8,5),5),5)BreatoxotaM(M(M(9,5),5),5)BreatodakaM(M(M(10,5),5),5)flexinegaM(2,6)OktiaM(2,7)flexitriaM(3,6)fainegaM(M(2,6),6)flexiteraM(4,6)funnynegaM(M(M(2,6),6),6)flexiperaM(5,6)ftetrinegaM(M(M(M(2,6),6),6),6)flexiectaM(6,6)fpentinegaM(M(M(M(M(2,6),6),6),6),6)flexizettaM(7,6)fhexinegaM(M(M(M(M(M(2,6),6),6),6),6),6)最后,Moser數(shù)=M(2,Mega-2)=M(2,M(2,3)-2),GrandMoser=M(3,GrandMega-2)=M(3,M(3,3)-2),GreatMoser=M(4,GreatMega-2)=M(4,M(4,3)-2),GongMoser=M(5,GongMega-2)=M(5,M(5,3)-2).【】箭頭時(shí)代【3,1】高德納上箭頭【3,1,1】定義及寫法高德納上箭頭符號(hào)一般用a↑c(diǎn)b表示,也可以連寫多個(gè)箭頭,如a↑↑↑b等.一般地,a↑↑…↑b(c個(gè)”↑”)=a↑c(diǎn)b.定義如下:a↑b=ab2.a↑c(diǎn)1=a3.a↑c(diǎn)b=a↑c(diǎn)-1(a↑c(diǎn)(b-1))(b>1,c>1)我們發(fā)現(xiàn),1條是乘ti,23ti和第四級(jí)運(yùn)算的定義幾乎完全相同.【3,1,2】性質(zhì)與聯(lián)系乘ti具有”右結(jié)合律”,a^b^c=a^(b^c),而這種箭頭的運(yùn)算也有同樣的性質(zhì),a↑nb↑nca↑n(b↑nc),而不是(a↑nb)↑nc.于是,我們得到:a↑n+1b=a↑na↑n…a(b個(gè)”a”).另外,第四級(jí)運(yùn)算可以劃歸為上箭頭的一類:ba=a↑↑b.注意到乘ti可以寫作a↑b,可以說,乘ti、第四級(jí)運(yùn)算和”↑”具有一種隱藏的聯(lián)系.實(shí)際上,通過對阿克曼函數(shù)的化歸,可以知道,Ack(a,b)=2↑a-2(b+3)-3(a>2)【3,1,3】其他記法a↑c(diǎn)ba^^…^b(c個(gè)”^”),a**…*b(c+1個(gè)”*”),或者a++…+b(c+2個(gè)”+”).如果說上箭頭是從乘ti開始的記號(hào),那么”**…*”就是從乘法開始的記號(hào),而”++…+”則是從加法開始的記號(hào).像上箭頭一樣,我們有多種表示”超運(yùn)算”的ti法.從加法開始的”超運(yùn)算”有下面這些hpenb),古德斯坦的Gnb)bn個(gè)南比爾的aXnb,nb(nbnb等.【3,2】葛立恒數(shù)【3,2,1】拉姆塞數(shù)把幾個(gè)點(diǎn)用一些線連起來,就形成了圖.如果一個(gè)圖中任意2個(gè)不同頂點(diǎn)之間剛好有一條線相連,就把它稱作完全圖.這里,nKn.現(xiàn)在,我們要對完全圖的邊進(jìn)行染色.拉姆塞數(shù)的定義如下:Kxn染色,滿足條件”不管如何染色,1種顏色的完全圖,2種顏色的完全圖,…,n種顏色的完全圖”x的最小值記作R(a1,a2,…,an).容易知道,R(1,a2,a3,…,an)=1,1K1.另外,R(m,2,2…,2)=m拉姆塞數(shù)還有對稱性,或者說交換律——比如R(4,5,6)=R(6,5,4)=R(6,4,5).事實(shí)上,拉姆塞數(shù)的計(jì)算是很困難的.到目前,已知的拉姆塞數(shù)只有很少.如:拉姆塞數(shù)可能取值拉姆塞數(shù)可能取值拉姆塞數(shù)可能取值R(3,3)6R(3,4)9R(3,5)14R(3,6)18R(4,17)[182,968]R(6,9)[169,780]R(3,7)23R(4,18)[182,1139]R(6,10)[179,1171]R(3,8)28R(4,19)[198,1329]R(6,11)[253,3002]R(3,9)36R(4,20)[230,1539]R(6,12)[262,4367]R(3,10)[40,43]R(4,21)[242,1770]R(6,13)[317,6187]R(3,11)[46,51]R(4,22)[282,2023]R(6,14)[317,8567]R(3,12)[52,59]R(5,5)[43,49]R(6,15)[401,11627]R(3,13)[59,69]R(5,6)[58,87]R(6,16)[434,15503]R(3,14)[66,78]R(5,7)[80,143]R(6,17)[548,20348]R(3,15)[73,88]R(5,8)[101,216]R(6,18)[614,26333]R(3,16)[79,135]R(5,9)[125,316]R(6,19)[710,33648]R(3,17)[92,152]R(5,10)[143,442]R(6,20)[878,42503]R(3,18)[98,170]R(5,11)[157,1000]R(6,21)[878,53129]R(3,19)[106,189]R(5,12)[181,1364]R(6,22)[1070,65779]R(3,20)[109,209]R(5,13)[205,1819]R(7,7)[205,540]R(3,21)[122,230]R(5,14)[233,2379]R(7,8)[216,1031]R(3,22)[125,252]R(5,15)[261,3059]R(7,9)[233,1713]R(3,23)[136,275]R(5,16)[278,3875]R(7,10)[232,2826]R(4,4)18R(5,17)[284,4844]R(7,11)[405,8007]R(4,5)25R(5,18)[284,5984]R(7,12)[416,12375]R(4,6)[35,41]R(5,19)[338,7314]R(7,13)[511,18563]R(4,7)[49,61]R(5,20)[380,8854]R(7,14)[511,27131]R(4,8)[56,84]R(5,21)[380,10625]R(7,15)[511,38759]R(4,9)[73,115]R(5,22)[422,12649]R(7,16)[511,54263]R(4,10)[92,149]R(5,23)[434,14949]R(7,17)[628,74612]R(4,11)[97,191]R(5,24)[434,17549]R(7,18)[722,100946]R(4,12)[128,238]R(5,25)[434,20474]R(7,19)[908,134595]R(4,13)[133,291]R(5,26)[464,23750]R(7,20)[908,177099]R(4,14)[141,349]R(6,6)[102,165]R(7,21)[1214,230229]R(4,15)[153,417]R(6,7)[113,298]R(8,8)[282,1870]R(4,16)[153,815]R(6,8)[127,495]R(8,9)[317,3583]R(8,10)[377,6090]R(9,9)[565,6588]R(8,11)[377,19447]R(9,10)[580,12677]R(8,12)[377,31823]R(10,10)[798,23556]R(8,13)[817,50387]R(11,11)[1597,184755]R(8,14)[817,77519]R(12,12)[1637,705431]R(8,15)[861,116279]R(13,13)[2557,2704155]R(8,16)[861,170543]R(14,14)[2989,10400599]R(8,17)[861,245156]R(15,15)[5485,40116599]R(8,18)[871,346103]R(16,16)[5605,155117519]R(8,19)[1054,480699]R(17,17)[8917,601080389]R(8,20)[1094,657799]R(18,18)[11005,2333606219]R(8,21)[1328,888029]R(19,19)[17885,9075135299]另外R(3,3,3)=17實(shí)際上,拉姆塞數(shù)有不少范圍,如

<R(k,k)≤4×R(k-2,k)+2,R(3,k)≤

,

e 2Cb 1 .12lnk

﹣a+b﹣2[注:本節(jié)內(nèi)容全為鋪墊]【3,2,2】葛立恒問題葛立恒提出這樣一個(gè)問題:n階超立ti體的任意兩個(gè)頂點(diǎn)都連一條線段,K2^n.對所有的線段進(jìn)行紅、藍(lán)二染色.求滿足條件”不管如何染色K4”n的最小值.M