Fibonacci
category: general [glöplog]
Fibonacci does your __int64s. Go make a demo about it.
the formula is this:
N = B*1.44042 + 2.1399
B = number of bits
N = number in which the overflow happens
B = 32 -> N = 48
B = 64 -> N = 94
B = 128 -> N = 186
N = B*1.44042 + 2.1399
B = number of bits
N = number in which the overflow happens
B = 32 -> N = 48
B = 64 -> N = 94
B = 128 -> N = 186
[0] 0
[1] 1
[2] 1
[3] 2
[4] 3
[5] 5
[6] 8
[7] 13
[8] 21
[9] 34
[10] 55
[11] 89
[12] 144
[13] 233
[14] 377
[15] 610
[16] 987
[17] 1597
[18] 2584
[19] 4181
[20] 6765
[21] 10946
[22] 17711
[23] 28657
[24] 46368
[25] 75025
[26] 121393
[27] 196418
[28] 317811
[29] 514229
[30] 832040
[31] 1346269
[32] 2178309
[33] 3524578
[34] 5702887
[35] 9227465
[36] 14930352
[37] 24157817
[38] 39088169
[39] 63245986
[40] 102334155
[41] 165580141
[42] 267914296
[43] 433494437
[44] 701408733
[45] 1134903170
[46] 1836311903
[47] 2971215073
[48] 4807526976
[49] 7778742049
[50] 12586269025
[51] 20365011074
[52] 32951280099
[53] 53316291173
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[55] 139583862445
[56] 225851433717
[57] 365435296162
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[59] 956722026041
[60] 1548008755920
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[434] 2244661544603472032436332649584708114319787957314032873538930901437280496774780497748874337
[435] 3631938672408255859550518752770954506578238315485816563618848933573305722729383091461442690
[436] 5876600217011727891986851402355662620898026272799849437157779835010586219504163589210317027
[437] 9508538889419983751537370155126617127476264588285666000776628768583891942233546680671759717
[438] 15385139106431711643524221557482279748374290861085515437934408603594478161737710269882076744
[439] 24893677995851695395061591712608896875850555449371181438711037372178370103971256950553836461
[440] 40278817102283407038585813270091176624224846310456696876645445975772848265708967220435913205
[441] 65172495098135102433647404982700073500075401759827878315356483347951218369680224170989749666
[442] 105451312200418509472233218252791250124300248070284575192001929323724066635389191391425662871
[443] 170623807298553611905880623235491323624375649830112453507358412671675285005069415562415412537
[444] 276075119498972121378113841488282573748675897900397028699360341995399351640458606953841075408
[445] 446698926797525733283994464723773897373051547730509482206718754667074636645528022516256487945
[446] 722774046296497854662108306212056471121727445630906510906079096662473988285986629470097563353
[447] 1169472973094023587946102770935830368494778993361415993112797851329548624931514651986354051298
[448] 1892247019390521442608211077147886839616506438992322504018876947992022613217501281456451614651
[449] 3061719992484545030554313848083717208111285432353738497131674799321571238149015933442805665949
[450] 4953967011875066473162524925231604047727791871346061001150551747313593851366517214899257280600
[451] 8015687004359611503716838773315321255839077303699799498282226546635165089515533148342062946549
[452] 12969654016234677976879363698546925303566869175045860499432778293948758940882050363241320227149
[453] 20985341020594289480596202471862246559405946478745659997715004840583924030397583511583383173698
[454] 33954995036828967457475566170409171862972815653791520497147783134532682971279633874824703400847
[455] 54940336057423256938071768642271418422378762132537180494862787975116607001677217386408086574545
[456] 88895331094252224395547334812680590285351577786328700992010571109649289972956851261232789975392
[457] 143835667151675481333619103454952008707730339918865881486873359084765896974634068647640876549937
[458] 232730998245927705729166438267632598993081917705194582478883930194415186947590919908873666525329
[459] 376566665397603187062785541722584607700812257624060463965757289279181083922224988556514543075266
[460] 609297663643530892791951979990217206693894175329255046444641219473596270869815908465388209600595
[461] 985864329041134079854737521712801814394706432953315510410398508752777354792040897021902752675861
[462] 1595161992684664972646689501703019021088600608282570556855039728226373625661856805487290962276456
[463] 2581026321725799052501427023415820835483307041235886067265438236979150980453897702509193714952317
[464] 4176188314410464025148116525118839856571907649518456624120477965205524606115754507996484677228773
[465] 6757214636136263077649543548534660692055214690754342691385916202184675586569652210505678392181090
[466] 10933402950546727102797660073653500548627122340272799315506394167390200192685406718502163069409863
[467] 17690617586682990180447203622188161240682337031027142006892310369574875779255058929007841461590953
[468] 28624020537229717283244863695841661789309459371299941322398704536965075971940465647510004531000816
[469] 46314638123912707463692067318029823029991796402327083329291014906539951751195524576517845992591769
[470] 74938658661142424746936931013871484819301255773627024651689719443505027723135990224027850523592585
[471] 121253296785055132210628998331901307849293052175954107980980734350044979474331514800545696516184354
[472] 196191955446197556957565929345772792668594307949581132632670453793550007197467505024573547039776939
[473] 317445252231252689168194927677674100517887360125535240613651188143594986671799019825119243555961293
[474] 513637207677450246125760857023446893186481668075116373246321641937144993869266524849692790595738232
[475] 831082459908702935293955784701120993704369028200651613859972830080739980541065544674812034151699525
[476] 1344719667586153181419716641724567886890850696275767987106294472017884974410332069524504824747437757
[477] 2175802127494856116713672426425688880595219724476419600966267302098624954951397614199316858899137282
[478] 3520521795081009298133389068150256767486070420752187588072561774116509929361729683723821683646575039
[479] 5696323922575865414847061494575945648081290145228607189038829076215134884313127297923138542545712321
[480] 9216845717656874712980450562726202415567360565980794777111390850331644813674856981646960226192287360
[481] 14913169640232740127827512057302148063648650711209401966150219926546779697987984279570098768737999681
[482] 24130015357889614840807962620028350479216011277190196743261610776878424511662841261217058994930287041
[483] 39043184998122354968635474677330498542864661988399598709411830703425204209650825540787157763668286722
[484] 63173200356011969809443437297358849022080673265589795452673441480303628721313666802004216758598573763
[485] 102216385354134324778078911974689347564945335253989394162085272183728832930964492342791374522266860485
[486] 165389585710146294587522349272048196587026008519579189614758713664032461652278159144795591280865434248
[487] 267605971064280619365601261246737544151971343773568583776843985847761294583242651487586965803132294733
[488] 432995556774426913953123610518785740738997352293147773391602699511793756235520810632382557083997728981
[489] 700601527838707533318724871765523284890968696066716357168446685359555050818763462119969522887130023714
[490] 1133597084613134447271848482284309025629966048359864130560049384871348807054284272752352079971127752695
[491] 1834198612451841980590573354049832310520934744426580487728496070230903857873047734872321602858257776409
[492] 2967795697064976427862421836334141336150900792786444618288545455102252664927332007624673682829385529104
[493] 4801994309516818408452995190383973646671835537213025106017041525333156522800379742496995285687643305513
[494] 7769790006581794836315417026718114982822736329999469724305586980435409187727711750121668968517028834617
[495] 12571784316098613244768412217102088629494571867212494830322628505768565710528091492618664254204672140130
[496] 20341574322680408081083829243820203612317308197211964554628215486203974898255803242740333222721700974747
[497] 32913358638779021325852241460922292241811880064424459384950843991972540608783894735358997476926373114877
[498] 53254932961459429406936070704742495854129188261636423939579059478176515507039697978099330699648074089624
[499] 86168291600238450732788312165664788095941068326060883324529903470149056115823592713458328176574447204501
[500] 139423224561697880139724382870407283950070256587697307264108962948325571622863290691557658876222521294125
Someone please make a graph out of the ratios of each consecutive number pair:
0,1
1,2
3,5
8,13
21,34
55,89
144,233
That's the real magic behind this pattern ;)
0,1
1,2
3,5
8,13
21,34
55,89
144,233
That's the real magic behind this pattern ;)
Quote:
Who is going for GMP now ?
Why use GMP when you can use SockZ?
Code:
1
0
1 ;f(n) f(n-1) n
.loop
COPY "f(" PRINT ")=" 3 OVER PRINT CR
3 SWAP ;n f(n-1) f(n)
SWAP2 ;n f(n) f(n-1)
OVER2 ;n f(n) f(n-1) f(n)
+ ;n f(n) f(n+1)=f(n-1)+f(n)
3 SWAP ;f(n+1) f(n) n
1 + ;f(n+1) f(n) n+1
COPY
502 SKIP= loop
END
And then:
f(500)=139423224561697880139724382870407283950070256587697307264108962948325571622863290691557658876222521294125
f(501)=225591516161936330872512695036072072046011324913758190588638866418474627738686883405015987052796968498626
Quote:
Go make a demo about it
I did ;D
I'll take your sweaty sockz and raise you with some erlang. (Obviously tail call recursive)
fibseq() ->
fibout(0,0),
fibseq(1,1,0).
fibseq(Num,Cur,Prev) when Num=<505 ->
fibout(Num,Cur),
Next=Cur+Prev,
fibseq(Num+1,Next,Cur);
fibseq(_,_,_) ->
done.
fibout(Num,Cur) ->
io:format("fib(~b)=~b~n",[Num,Cur]).
---------------------------------------------------------
46> test:fibseq().
fib(0)=0
fib(1)=1
fib(2)=1
fib(3)=2
fib(4)=3
fib(5)=5
........
........
fib(501)=225591516161936330872512695036072072046011324913758190588638866418474627738686883405015987052796968498626
fib(502)=365014740723634211012237077906479355996081581501455497852747829366800199361550174096573645929019489792751
fib(503)=590606256885570541884749772942551428042092906415213688441386695785274827100237057501589632981816458291377
fib(504)=955620997609204752896986850849030784038174487916669186294134525152075026461787231598163278910835948084128
fib(505)=1546227254494775294781736623791582212080267394331882874735521220937349853562024289099752911892652406375505
done
47>
fibseq() ->
fibout(0,0),
fibseq(1,1,0).
fibseq(Num,Cur,Prev) when Num=<505 ->
fibout(Num,Cur),
Next=Cur+Prev,
fibseq(Num+1,Next,Cur);
fibseq(_,_,_) ->
done.
fibout(Num,Cur) ->
io:format("fib(~b)=~b~n",[Num,Cur]).
---------------------------------------------------------
46> test:fibseq().
fib(0)=0
fib(1)=1
fib(2)=1
fib(3)=2
fib(4)=3
fib(5)=5
........
........
fib(501)=225591516161936330872512695036072072046011324913758190588638866418474627738686883405015987052796968498626
fib(502)=365014740723634211012237077906479355996081581501455497852747829366800199361550174096573645929019489792751
fib(503)=590606256885570541884749772942551428042092906415213688441386695785274827100237057501589632981816458291377
fib(504)=955620997609204752896986850849030784038174487916669186294134525152075026461787231598163278910835948084128
fib(505)=1546227254494775294781736623791582212080267394331882874735521220937349853562024289099752911892652406375505
done
47>
Hit me with some bbcode to get the erlang code to look right.
Code:
fibseq() ->
fibout(0,0),
fibseq(1,1,0).
fibseq(Num,Cur,Prev) when Num=<505 ->
fibout(Num,Cur),
Next=Cur+Prev,
fibseq(Num+1,Next,Cur);
fibseq(_,_,_) ->
done.
fibout(Num,Cur) ->
io:format("fib(~b)=~b~n",[Num,Cur]).
bah, humbug
mov ch, 2
mov di, cx
rep stosb
inc ax
mov ch, 120
loop1:
stosb
mov al, [di-160]
adc al, [di-80]
aaa
loop loop1
mov ch, 120
loop2:
dec di
mov dl, [di]
add dl, '0'
mov ah, 2
int 21h
loop loop2
ret
mov ch, 2
mov di, cx
rep stosb
inc ax
mov ch, 120
loop1:
stosb
mov al, [di-160]
adc al, [di-80]
aaa
loop loop1
mov ch, 120
loop2:
dec di
mov dl, [di]
add dl, '0'
mov ah, 2
int 21h
loop loop2
ret
funny trick.. tmdc next?
Skate pwns.
8bit always wins ;)
Code:
def number2bin(x):
out = []
while x:
out.append( x%2 )
x /= 2
out.reverse()
return out
def jump( p, c ):
return p*p+c*c, (2*p+c)*c
def step( p, c ):
return c, p+c
def fibTravel(n):
marsenNumber = 2**4423-1 #quite large finite field
path = number2bin(n)
p,c = 1,0 #starting point
for i in path[:-1]:
#print "wow %s"%i
if i: p,c = step( p,c )
p,c = jump( p,c )
p = p % marsenNumber
c = c % marsenNumber
if path[-1]: p,c = step( p,c )
return c
f[506]=2501848252103980047678723474640612996118441882248552061029655746089424880023811520697916190803488354459633
f[507]=4048075506598755342460460098432195208198709276580434935765176967026774733585835809797669102696140760835138
f[508]=6549923758702735390139183573072808204317151158828986996794832713116199613609647330495585293499629115294771
and now... PANTS OFF (klękać do miecza)
f[47239817523981479281579218712460998291842112345432721649875243104932184651324] mod 2^4423 -1 =
124213441849311985446486018309175024933475033276087063062089696248033101910610427635944517630958231376181716158561902062371516742409767758834435441405570893269249045060256125990371877912044148825319792686909561776306751418472751588433411263723013321390378299378413993894937287864425061640881684362299476227148562973478775910386785676258243180956427734699057184286328519447284384469511322517622057287637919668717996977676163388867902453438587927215439632731538997024220685467385937038336095682959238648592241561516284581604489249622627216449129369061099892823550416514323544641692378708504640765043123163318928255548543957067332112463484216126702609406073311143762026062827561146120396373940437387398678415643698370242621975322052191638544935846743517542902304555879808794367931137441796327573084270351315207131292438782293490185898684059165752267235422780507663341695583588502263319754618902661335400774394074803945936754341743709054774672091572058751408911933004591460552901569464354487149619654667716213162432851863195378417417600569604094987479437247031071156110463537773293271210539186992574788335622957102826912974617335683942840251069600758338551361399313453525280113905544366649972992200703001732891903378470158218905574164778990136504502389519632463305011739190433881567910845452607508467238819291659562069131629834756078711
time computing < 1[s] :)
Ok, lets play with Fibonacci numbers and... intros 256b. :)
Intro 256 is just a sequence of bits, isn't it? - so, it could be turn into natural number. Quick and easy code:
I took famous "puls.com" (by Rrrola), and this is its 'cardinal' number:
N=24690664190877577863735026143204829428400636123616184013657436330923929759807169312891548985841320838430606346600701597160176067658946940952081310487644041301399425716762418092371943992503766295923098750083587105426035587821110314006728025461173822613996696154500985926597060896584077024948909961013674874687933405515561348255854377601447793234032598684705213532461873882287131900652785679897636660207612839176782238414004955599453527014398653898669331941852252053786290887760870160484763311038944231731218062374572171434998015433563299845892454555749208637453597117970206952510950529863791439474064675977657075307440
And this is N'th Fibonacci number:
F[N] mod 2^4423-1= 249349349797732314358551559033855664958972652807101321853717325428222442911180916905942322999775721778569959454821189325432976614689834413352442505725887212940869379972449730472843212244846846359391447505803445262075294244323721204339373013535239988015766355010923416879351136789393617315197620047147993987661324807264215557813667859344792337426617615799832425616817647605010483062934675447046400940627349014219808252042326080771315042637147928111096210538032962615571575488998909640503876815636732057727673712290181872687981444075318315196048953422139464899558810018747378161128600616589080589046827649894520783449062553368802246840999872266221732401094009483353951684977538862878157216289520665952330438861977150901612148980960074778754716816620128599529600384488466344199792584686150943332014913655690480055395646925270183205618635881567153235346177483753932319237068794106550743326310472900442200976021302629880126165820208805273508699554081700320287984246077819565231287660119565023980125109649821779019270131086768604833136342322826503114003866468651608244157944803243737449221209460500736325145846349585292855968904099856064929512547098293618926362440584649881369288432947330490987740802498376395277905692060462819060116891667801828525471606152794615330607612147833699202118470905604162235159752114477201234865669171370024546
Or.. we can note it as follows
F(
) mod 2^4423-1 =
249349349797732314358551559033855664958972652807101321853717325428222442911180916905942322999775721778569959454821189325432976614689834413352442505725887212940869379972449730472843212244846846359391447505803445262075294244323721204339373013535239988015766355010923416879351136789393617315197620047147993987661324807264215557813667859344792337426617615799832425616817647605010483062934675447046400940627349014219808252042326080771315042637147928111096210538032962615571575488998909640503876815636732057727673712290181872687981444075318315196048953422139464899558810018747378161128600616589080589046827649894520783449062553368802246840999872266221732401094009483353951684977538862878157216289520665952330438861977150901612148980960074778754716816620128599529600384488466344199792584686150943332014913655690480055395646925270183205618635881567153235346177483753932319237068794106550743326310472900442200976021302629880126165820208805273508699554081700320287984246077819565231287660119565023980125109649821779019270131086768604833136342322826503114003866468651608244157944803243737449221209460500736325145846349585292855968904099856064929512547098293618926362440584649881369288432947330490987740802498376395277905692060462819060116891667801828525471606152794615330607612147833699202118470905604162235159752114477201234865669171370024546
So... equation:
F[ intro256 ] mod q = x
is easy to solve... what about
F[x] mod q = intro256 ? :)
Another question is, how set q number (finite filed)? I have just taken 2**4423-1, which is prime number.
Aha, there is one mistake in my fibTravel function - there should be:
bye!
Intro 256 is just a sequence of bits, isn't it? - so, it could be turn into natural number. Quick and easy code:
Code:
import fib #his is the file from my last post
def introNumber():
f = open("puls.com","rb") #yeah.. this is THIS pulse 256b intro
try:
byte = f.read(1)
index=0
sum=0
while byte != "":
val = int(byte.encode('hex'), 16)
sum += val*(256**index)
byte = f.read(1)
index += 1
finally:
f.close()
return sum
print introNumber()
print "--------------------------"
print fib.fibTravel( introNumber() )
I took famous "puls.com" (by Rrrola), and this is its 'cardinal' number:
N=24690664190877577863735026143204829428400636123616184013657436330923929759807169312891548985841320838430606346600701597160176067658946940952081310487644041301399425716762418092371943992503766295923098750083587105426035587821110314006728025461173822613996696154500985926597060896584077024948909961013674874687933405515561348255854377601447793234032598684705213532461873882287131900652785679897636660207612839176782238414004955599453527014398653898669331941852252053786290887760870160484763311038944231731218062374572171434998015433563299845892454555749208637453597117970206952510950529863791439474064675977657075307440
And this is N'th Fibonacci number:
F[N] mod 2^4423-1= 249349349797732314358551559033855664958972652807101321853717325428222442911180916905942322999775721778569959454821189325432976614689834413352442505725887212940869379972449730472843212244846846359391447505803445262075294244323721204339373013535239988015766355010923416879351136789393617315197620047147993987661324807264215557813667859344792337426617615799832425616817647605010483062934675447046400940627349014219808252042326080771315042637147928111096210538032962615571575488998909640503876815636732057727673712290181872687981444075318315196048953422139464899558810018747378161128600616589080589046827649894520783449062553368802246840999872266221732401094009483353951684977538862878157216289520665952330438861977150901612148980960074778754716816620128599529600384488466344199792584686150943332014913655690480055395646925270183205618635881567153235346177483753932319237068794106550743326310472900442200976021302629880126165820208805273508699554081700320287984246077819565231287660119565023980125109649821779019270131086768604833136342322826503114003866468651608244157944803243737449221209460500736325145846349585292855968904099856064929512547098293618926362440584649881369288432947330490987740802498376395277905692060462819060116891667801828525471606152794615330607612147833699202118470905604162235159752114477201234865669171370024546
Or.. we can note it as follows
F(
) mod 2^4423-1 =249349349797732314358551559033855664958972652807101321853717325428222442911180916905942322999775721778569959454821189325432976614689834413352442505725887212940869379972449730472843212244846846359391447505803445262075294244323721204339373013535239988015766355010923416879351136789393617315197620047147993987661324807264215557813667859344792337426617615799832425616817647605010483062934675447046400940627349014219808252042326080771315042637147928111096210538032962615571575488998909640503876815636732057727673712290181872687981444075318315196048953422139464899558810018747378161128600616589080589046827649894520783449062553368802246840999872266221732401094009483353951684977538862878157216289520665952330438861977150901612148980960074778754716816620128599529600384488466344199792584686150943332014913655690480055395646925270183205618635881567153235346177483753932319237068794106550743326310472900442200976021302629880126165820208805273508699554081700320287984246077819565231287660119565023980125109649821779019270131086768604833136342322826503114003866468651608244157944803243737449221209460500736325145846349585292855968904099856064929512547098293618926362440584649881369288432947330490987740802498376395277905692060462819060116891667801828525471606152794615330607612147833699202118470905604162235159752114477201234865669171370024546
So... equation:
F[ intro256 ] mod q = x
is easy to solve... what about
F[x] mod q = intro256 ? :)
Another question is, how set q number (finite filed)? I have just taken 2**4423-1, which is prime number.
Aha, there is one mistake in my fibTravel function - there should be:
Code:
if path[-1]: p,c = step( p,c )
p = p % marsenNumber
c = c % marsenNumber
return c
bye!
Quote:
So... equation:
F[ intro256 ] mod q = x
is easy to solve... what about
F[x] mod q = intro256 ? :)
We are not sure that g(x)=(fib(x) mod q) is an injection (and i see no simple reason why it would be so). We shall check if there exist two numbers N, M such that g(N)=g(M)...
Also in the second part, shall not q be 2^256 to have a 256b intro as a result?
Quote:
Also in the second part, shall not q be 2^256 to have a 256b intro as a result?
Yeah, if want 256b intro it should be 256^256 = 2^8*256 =2^2048 but... I think that it will be better if it will be prime number. Ex.16'th Mearsenn number 2^2203-1
http://en.wikipedia.org/wiki/Mersenne_prime
Why? I don't know - but if there is prime number, every formula is smarter. :P
And one small point. If q is prime, it is called 'finite filed' and there is a lot of webpages about its properties
'finite field'. Once is a typo, twice a coincidence?
fib(x) mod 11 repeats the sequence 1,1,2,3,5,8,2,10,1,0. There's no solution for 4, 6, 7 and 9. Considering that 11 is a prime I doubt 2^2203-1 works any better than 2^2048.
fib(x) mod 11 repeats the sequence 1,1,2,3,5,8,2,10,1,0. There's no solution for 4, 6, 7 and 9. Considering that 11 is a prime I doubt 2^2203-1 works any better than 2^2048.
Quote:
...should be 256^256 = 2^8*256 =2^2048
Ooops... You're right, i forgot a byte has 8 bits! Shame on me! ;p
Quote:
'finite field'. Once is a typo, twice a coincidence?
It was just a typo - finite field is correct of course
Quote:
fib(x) mod 11 repeats the sequence 1,1,2,3,5,8,2,10,1,0. There's no solution for 4, 6, 7 and 9. Considering that 11 is a prime I doubt 2^2203-1 works any better than 2^2048.
Yes, it is absolutely true. I found that such sequences are called Pisano periods
But...
On the other hand, not all numbers from 0...256^256 are properly constructed binaries for x86