Human being is a living random number generator. Abstract: General wisdom is, mathematical operation is needed to generate number by numbers.

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1 Huan beng s a lvng o nube geneato Anda Mta Anushat Abasan, Utta halgun -7, /AF, alt Lae, olata, West Bengal, 764, Inda Abstat: Geneal wsdo s, atheatal oeaton s needed to geneate nube by nubes It s onted out that wthout any atheatal oeaton tue o nubes an be geneated by nubes though algoth oess It les that huan ban tsel s a lvng tue o nube geneato Huan ban an eet the enoous huan de o tue o nubes Nube as well as a stng o non/seudo-o nubes an be geneated by nubes wth the hel o atheatal oeaton though algoth oess [,] Although thee s no atheatal oo that o nubes annot be geneated by any lassal algoth, but t s wdely beleved that o nubes an be geneated only by usng o events It ay be ealled that von Neuann st uled out [] the ossblty o geneatng o nubes by any atheatal oeaton Thee s a aveat von Neuann dd not ule out the geneaton o o nubes by any algoth tehnque In at, t s ossble to geneate o nubes by usng sle quantu algoth [] whh exlots ntns oness o quantu syste Fo the above dsusson t ay be onjetued that any lassal algoth does not eque atheatal oeaton to geneate nubes, then t an be a good date o the geneaton o both o non-o nubes We have ound suh algoth Let us st esent the ost sle one

2 Algoth- I a) Tae a stng o n o bts b)note down the o ostons o s n the asendng ode It gves a sequene o ostve nteges laly, note down the o ostons o s n the asendng ode It gves anothe sequene o ostve nteges The oeaton an be sybolally desbed as, ) Tae anothe sequene o n o bts, d) Aly on, theeby extat a a o sequenes o o bts It eans bts o esetvely That s, bts o ae extated o soe o ostons n ae extated o those ostons n bts o ae extated o those bt ostons n The oeaton an be sybolally desbed as : :: detened by whh ae the ostons o s n whh ae the ostons o s n : :: In ths oeaton, two bts bb ae used to geneate a sngle bt b whee the st bt belongs to seond bt belongs to Fo,, Fo,, the It s tval to see that the algoth geneates a bt o wth obablty / oo: ne s a sequene o o bts, a o obablty o gettng o n any oston s / ne s a sequene o o bts, a o obablty o gettng o n any oston s / ne ae unoelated ndeendent sequenes ae ndeendent obabltes The obablty o geneatng a bt o, whh wll onstut, asng o two ndeendent obabltes s /4

3 The obablty o geneatng a bt o, whh wll onstut, asng o two ndeendent obabltes s /4 The total obablty o geneatng a bt o whh wll onstut o + / s The geneated bts ae o bts Gven 4n bts n the nut n bts ae geneated n the outut Gan s nsgnant To nease the gan, we use the geneated bts as one o the nut euse one o the gven nut, then the geneated bts wll not be o bts beause two nuts (n+n) wll not be ndeendent nuts It does not ean that the geneated bts annot be uthe used to geneate bts Laws o natue tells that that sngle geneaton s ossble, then nnte geneaton s also ossble Thee s no exeton to ths natual law Next we shall esent a ototye o the above algoth whh an geneate abtay nubes o o bts te # Algoth- II a) Tae a sequene b) Note down the o ostons o s s n the asendng ode It gves two sequenes, o ostve nteges The oeaton an be sybolally desbed as, ) Tae anothe sequene d) onstut oleentay sequene o, by ntehangng the ostons o Note that ae axally oelated o sequenes These two sequenes have to be used at o e) Tae anothe sequene o o bts In -th ste, o wll be hosen by the -th bt o

4 te # a) hoose o by the st bt o b) Aly on the hosen o, theeby extat a a o sequenes o o bts The oeaton an be desbed as ) Attah : ( / ) ::, theeby onstut a new sequene : ( / ) :: o n o bts Hee attahent eans s ollowed by That s, -th bt o s also the -th bt o the attahed stng The attahent an be sybolally desbed as d) Note down the ostons o s s n It gves a new a o sequenes o ostve nteges n the asendng ode Note down the ostons o s s n the stng The oeaton an be sybolally desbed as te # a) hoose o by the nd bt o b) Aly on the hosen o, theeby extat a new a o sequenes The oeaton an be desbed as : ( / ) :: : ( / ) :: ) Attah, theeby onstut a new n-bt stng The attahent an be sybolally desbed as d) Note down the ostons o s s n It gves a new a o sequenes ostve nteges The oeaton an be sybolally desbed as o

5 te # a) hoose o by the -th bt o b) Aly (taen o evous ) on the hosen o, extat a new a o bt stngs To get, bts ae extated o those bt ostons n ethe o whee bts ae s n To get, bts ae extated o those bt ostons n ethe o whee bts ae s n The oeatons an be sybolally desbed as : ( / ) :: : ( / ) :: ) Attah, theeby onstut a new n-bt stng The attahent an be sybolally desbed as d) Note down the ostons o s s n It gves a new a o sequenes, ostve nteges The oeaton an be sybolally desbed as, o uose, n -th ste the ntege sequenes wll be aled on bt o s Let us llustate the algoth stes -th bt o s o on -th te # a) uose the ollowng s taen b) {,, 5, 6, 8,, 4,5 } {, 4, 6,7, 9,,,,6 } ) uose the ollowng s taen d)

6 e) uose the ollowng s taen { } te # a) hoose as st bt o s b) Aly on ) Attah The attahed sequene s, { } d) The extated ntege stngs ae {,, 4, 5,,, 5, 6} {, 6, 7, 8, 9,,, 4} te: # a) hoose as nd bt o s b) Aly on ) Attah The attahed sequene s, { }

7 d) The extated ntege stngs ae {,, 7,,, 4, 5, 6} {, 4, 5, 6, 8, 9,, } Let us now ove that n eah ste the algoth-ii geneates o bts oo : ne / ne oston s s a sequene o o bts a o obablty o gettng o n j-th oston s s olentay to, a o obablty o gettng o n the sae j-th / Note that ae unoelated ndeendent sequenes laly, ae unoelated ndeendent sequenes In eah ste, o obablty / Theeoe, total obablty o gettng o n any oston s, + / s hosen by a bt o wth ne s a sequene o o bts, a o obablty o gettng o s / ne ae unoelated ndeendent sequenes geneatng a bt o, whh wll onstut ae ndeendent obabltes The obablty o, asng o two ndeendent obabltes /4 s The obablty o geneatng a bt o, whh wll onstut, asng o two ndeendent obabltes s /4 The total obablty o geneatng a bt o whh wll onstut + / o s ne extated a onstuts, an be denoted as ne ae unoelated ndeendent sequenes, whh wll onstut ae ndeendent obabltes The obablty o geneatng a bt o, asng o two ndeendent obabltes /4 s

8 The obablty o geneatng a bt o, whh wll onstut, asng o two ndeendent obabltes s /4 The total obablty o geneatng a bt o whh wll onstut + / o s, ne extated a o (-)-th ste onstuts, an be denoted as ne ae unoelated ndeendent sequenes ae ndeendent obabltes The obablty o geneatng a bt o, whh wll onstut, nvolvng two ndeendent obabltes /4 s The obablty o geneatng a bt o, whh wll onstut, nvolvng two ndeendent obabltes s /4 The total obablty o geneatng a bt o whh wll onstut + In eah ste algoth geneates o bts / s, oe o the eatues o the algoth ae dsussed below An attahed stng an beoe dental wth o dental wth o o I any attahed stng beoes the algoth wll halt I attahed stng beoes dental wth the algoth wll be yl The obablty o geneatng dental o o s / n In R stes the total nube o geneated o bts s nr whle the nube o nut o bts s 4n + R Gan s, G nn (4n +R) oe o the geneated o bts an be used to nease the length o the nut stngs stngs ( ), The algoth an ultately un on two abtaly long

9 As a as oelaton s onened thee ae anly two ossbltes a) In eah ste, ndeendent nuts ae gven thee s no oelaton between the nuts o -th stes j-th stes In ths ase, -th nut an be oelated wth the -th outut Ths s not n-bt n-bt oelaton Ths s n- bt n-bt oelaton But -th outut annot be oelated wth j-th outut -th nut annot be oelated wth j-th outut o n-bt oututs an ean unoelated even they ae oelated wth nbt nuts b) In eah ste, ndeendent nuts ae gven but thee s a oelaton between the nuts o -th stes j-th stes The algoth II s an exale In ths ase, -th nut an be oelated wth the -th outut -th nut annot be oelated wth the j-th outut, but -outut annot be oelated wth j-th outut But ths oelaton an be nown only the nut stngs ae nown Theeoe, so a outut s onened ths s a eanngless oelaton Due to ths eanngless oelaton the sae outut an be geneated by gvng the sae nut Ths s advantageous o Monte-alo sulaton when sulaton needs to be eeated unde the sae ondton One ay tell t as eanngully eanngless oelaton 4 In the outut the obable nube o deent n-bt sequenes s n Fo all obable nut as the obable nube o deent ode o n-bt sequenes o sequene s atula nuts n n Fo a a o the obable nube o deent ode o n-bt sequenes o sequene s n A a o attahed stngs an be used as a new a o nuts Theeoe, statng wth ntal a n all obable n sequenes o sequenes an be geneated The obablty o geneatng any n-bt sequene s / n The obablty o geneatng the sae nbt sequene s also / n The obablty o geneatng any sequene o n-bt sequenes s

10 The sae outut sequenes an be geneated the sae nuts ae gven Ths s n n advantageous Ro events annot geneate the sae dstbuton we de so 5 Gven the a o outut sequene t s ossble to denty the a o nut sequenes Beause the n sae a o outut sequenes an be geneated by as o nut sequenes But the queston s, gven the sequene o oututs o the algoth s t ossble to denty the algoth? Ths s not ossble we an nd anothe ototye o ths algoth One o the ototye o the esented algoth s dsussed below Instead o one sequene two sequenes an be sultaneously aled ethe on o at o to geneate ou sequenes o o bts tang the bts o those ostons o o whee bts ae : ) s n s n ) s n s n ) s n s n 4) s n s n Total obablty o geneatng a bt wll be /8 +/8 +/8 + /8 / As thee ons jontly geneate a bt wth obablty / We onjetue that exonentally lage nubes o ototyes o ths algoth do exst an be ound by exonentally lage nubes o eseahes 6 The algoth an also be used to geneate non-o nubes I one o the thee nut sequenes ontans non-o nubes the algoth wll geneate non-o nubes The algoth geneates sequene o two equally obable nubes X whee, Followng the sae algoth tehnque sequene o N equally obable nubes X an be geneated whee,, N In eah ste, N sequenes, eah onssts o N equally obable nubes X, an be geneated evously geneated N sequenes ae attahed then the attahed sequene o length Nn s aled on any one o the N! olentay sequenes,,,, eah o length Nn, at o It

11 aeas that n eah ste, to selet N! sequenes (at o) wth equal obablty /N! we need a sequene o N! equally obable nubes But the oble s, we need an abtaly long sequene o N! equally obable nubes to geneate abtaly long sequene o N equally obable nubes Ths s not an nsuountable oble We have a ound a soluton whh s dsussed below The equeent o abtaly long sequene o N! equally obable nubes an be avoded we st geneate sequene o equally obable nubes ollowed by sequene o equally N obable nubes nally sequene o (N-) equally obable nubes by ths algoth tehnque Along wth ths sequenes anothe sequene N o N equally obable nubes an be N taen Now we tae the -th nubes o the sequenes, N,, then N! dstbutons ae ossble Eah aangeent (X ) (X ) (X 4 ) (X N ) an be used to hoose any one o the sequenes,,, N! wth equal obablty /N! O ouse whh sequene wll be hosen by whh obable dstbuton that nstuton should be gven n the nut To geneate a sequene o N obable nubes we have to geneate sequenes o st, then nally N- equally obable nubes Geneatng,, 4 N equally obable nubes they an be used to nease the length o the sequenes, N, esetvely Genealzed Algoth te # a) Tae a sequene o N equally obable nubes X whee the length o s Nn b) Note down the o ostons o X, X, X X n the asendng ode whh gves sequenes,, o ostve nteges The oeaton an be desbed as,, ) Tae anothe sequene d) Fo o N equally obable nubes X whee the length o, onstut olentay sequenes,, o s Nn by ntehangng the ostons o X ne N nubes an be aanged n N! ways so ostons o N nubes an be ntehanged n N!

12 ways Theeoe, N! olentay sequenes o N equally obable nubes ae ossble I one o these olentay sequenes s nown then the othes an be onstuted e) Tae a set o sequenes N (,, ) o,, (N-), N obable nubes te # a) hoose o o o N o the sequenes,,, by the aangeent (X ) (X ) (X 4 ) (X N ) o -th nube o eah b) Aly -, - - (taen o evous ) on the hosen o o o extat the sequenes, The oeaton an be sybolally desbed as : ( / / / ) :: : ( / / / ) :: : ( / / / ) :: - In ths oeaton two nubes XX ae used to geneate a sngle nube X whee the st nube X belongs to the seond nube X belongs to The odng an be desbed as Fo, X X X, X X X,, X X X Fo, X X X, X X X,, X X X Fo, X X X, X X X,, X X X ) Attah,, theeby onstut a new sequene The oeaton an be sybolally desbed as -

13 d) Note down the ostons o X, X, X X n It gves a new sequenes,, o ostve nteges The oeaton an be sybolally desbed as,, As o exale, o the geneaton o abtaly long stng o equally obable nubes we need nuts:, {,,, 4,5, 6 }, {, } Theeoe, we have to st geneate abtaly long sequene o equally obable nubes to geneate abtaly long sequene o equally obable nubes by ths algoth Hee, out o! aangeents one aangeent (X ) (X ) o -th nubes o wll detene one o the sx sequenes,,,,, on whh be aled wth obablty /6 Note that equally obable nubes ae the ay seeds wll The obablty o geneatng any sequene obablty o geneatng the sae sequene s also sequene o sequenes s o N equally obable nubes X s Nn N Nn N The The obablty o geneatng any Geneated stngs annot be oelated aong theselves N Nn N Nn Gven a stng o o nube t s not ossble to tell how t was geneated Le evous ase, t s easy to see that ths genealzed algoth geneates a nube wth obablty /N oo : The obablty o seletng any one o the N! sequenes,,, N! by the aangeent N o -th nubes o the sequenes,,, s ne s 4 N N! sequene o o nubes a o obablty o gettng a nube X n j-th oston s /N ne s a olentay sequene o a o obablty o gettng X n the sae j-th oston o

14 the sequene s /N ne gettng X N n the sae j-th oston o the sequene N! s a olentay sequene o a o obablty o N! s N! /N Note that the set o sequenes N (,,, ) s ndeendent o the set o sequenes (,,, N! ) In eah ste, one sequene s hosen by the o nube o wth obablty /N Total obablty o gettng a nube X n any oston s, N! ( N! ) N! /N (/N!/N +/N!/N +/N! /N /N!/N) ne s a sequene o o nubes a o obablty o gettng eah nube X n s /N ne ae unoelated ndeendent sequenes ae ndeendent obabltes The obablty o geneatng a nube, whh wll onstut, asng o two ndeendent obabltes s N / N Total obablty o geneatng a nube X whh wll onstut o o o s, + + N N (/N + /N + +/N ) /N Ro nubes ae extensvely used n Monte alo sulaton o ytogah uoses [] The algoth an be used o these uoses The algoth an be used to geneate unbeaable essage, IN asswod Even oon eole an use the algoth o these uoses The ytogah use o the algoth has been dsussed [4,5] elsewhee The algoths eveals that thee s a nueal ogn o oness Needless to say, the algoths an be leented on lassal oute The exstene o these algoths les that lassal

15 oute huan ban an behave le a obablst ahne It ay be ealled that Feynan envsoned [6] quantu oute o the sulaton o quantu syste Although lassal oute s not sutable o the sulaton o quantu syste, but ntns oness o quantu syste an be sulated on lassal oute The exstene o obablst algoths [7,8] nludng Monte-alo based algoths [] gave hnt that lassal oute annot be onsdeed as a totally detenst ahne The esented algoth ooboates that obsevaton Note added: Inteested eades ay see soe wely elated wos [9-] eal: tan@yahooo Reeenes J von Neuann,96 Vaous Tehnques Used n onneton Wth Ro Dgts John von Neuann, olleted Wos AH Taub, ed, MaMllan D E nuth, The at o oute ogang Vol e nueal Algoths ublshe - Addson- Wesley,99 M A Nelsen I huang, Quantu outaton Quantu Inoaton, nd edton, abdge unvesty ess, 4 A Mta, An unbeaable ytosyste, htt://xxxlanlgov/sr/499 5 A Mta, An unbeaable ytosyste o oon eole, htt://xxxlanlgov/sr/65 6 R Feynan, Int J Theo hy, 467, (98) 7 R olovay Vtassen, IAM J out, 6, 84,(976) 8 MORabn, J Nube theoy,,8, (98) ************************************************************************** 9 AMta, Quantu ehans an alone ovde seuty n ounaton,htt://xxxlanlgov/sit/5 A Mta, Honesty an be the best oly wthn quantu ehans, htt://xxxlanlgov/sr/5859 A Mta, Quantu theoy an be olletvely veed,, htt://xxxlanlgov/s,r/56 A Mta, A atle an ay oe than one bt o noaton, htt://xxxlanlgov/sit/57

GENERALIZATION OF AN IDENTITY INVOLVING THE GENERALIZED FIBONACCI NUMBERS AND ITS APPLICATIONS

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