dan73
Posts:
468
From:
ct
Registered:
6/14/08


Re: 18 consecutive zeros in a power of two
Posted:
Oct 1, 2012 12:35 AM


> I have found the first power of two which contains a > run of 18 > consecutive zeros in its decimal form. It is > 2^4,627,233,991. > > For information on the algorithm that I used, see: > http://mathforum.org/kb/thread.jspa?messageID=7769808 > > I thank Christian Bau for his "multiply by 2^34" > method which is much > faster than the "multiply by 2^11" method that I had > been using. > > For the The OnLine Encyclopedia of Integer Sequences > entry, see: > https://oeis.org/A006889 > > 2^4,627,233,991 (a 1,392,936,229 digit number) > contains the sequence > "813000000000000000000538" about 99.5% of the way > through. The > computation took about six months on a PC with 8 > processors. The > program was written in C#. > > Counting from the most significant digit we have... > > 1,385,589,639 digits > 18 zeros > 7,346,572 digits >  > 1,392,936,229 total length > > Below, are corresponding sections of the digit string > from > 2^4,627,233,950 to 2^4,627,233,991. > > The program reached 2^4,627,233,954. It did not find > a run of eleven > zeros anywhere so it multiplied by 2^34, taking it to > 2^4,627,233,988. > Here it found a string of 16 zeros, with a "5" to the > left of them  > so it just doubled the number. > Then it found a string of 16 zeros, with a "5" to the > left of them  > so it just doubled the number. > Then it found a string of 17 zeros, with a "5" to the > left of them  > so it just doubled the number. > It was now at 2^4,627,233,991 and stopped, having > found the 18 zeros. > > 4,627,233,950: 51961535532592194158496567979454994  > 4,627,233,951: 03923071065184388316993135958909988  > 4,627,233,952: 07846142130368776633986271917819977  > 4,627,233,953: 15692284260737553267972543835639955  > 4,627,233,954: 31384568521475106535945087671279911  > << x2^34 > 4,627,233,955: 62769137042950213071890175342559822  > 4,627,233,956: 25538274085900426143780350685119644  > 4,627,233,957: 51076548171800852287560701370239289  > 4,627,233,958: 02153096343601704575121402740478578  > 4,627,233,959: 04306192687203409150242805480957156  > 4,627,233,960: 08612385374406818300485610961914313  > 4,627,233,961: 17224770748813636600971221923828626  > 4,627,233,962: 34449541497627273201942443847657253  > 4,627,233,963: 68899082995254546403884887695314506  > 4,627,233,964: 37798165990509092807769775390629012  > 4,627,233,965: 75596331981018185615539550781258024  > 4,627,233,966: 51192663962036371231079101562516049  > 4,627,233,967: 02385327924072742462158203125032099 1 > 4,627,233,968: 04770655848145484924316406250064198 2 > 4,627,233,969: 09541311696290969848632812500128396 2 > 4,627,233,970: 19082623392581939697265625000256792 3 > 4,627,233,971: 38165246785163879394531250000513584 4 > 4,627,233,972: 76330493570327758789062500001027169 4 > 4,627,233,973: 52660987140655517578125000002054338 5 > 4,627,233,974: 05321974281311035156250000004108676 6 > 4,627,233,975: 10643948562622070312500000008217353 7 > 4,627,233,976: 21287897125244140625000000016434707 7 > 4,627,233,977: 42575794250488281250000000032869415 8 > 4,627,233,978: 85151588500976562500000000065738830 9 > 4,627,233,979: 70303177001953125000000000131477661 9 > 4,627,233,980: 40606354003906250000000000262955323 10 > 4,627,233,981: 81212708007812500000000000525910647 11 > 4,627,233,982: 62425416015625000000000001051821294 11 > 4,627,233,983: 24850832031250000000000002103642589 12 > 4,627,233,984: 49701664062500000000000004207285179 13 > 4,627,233,985: 99403328125000000000000008414570358 14 > 4,627,233,986: 98806656250000000000000016829140716 14 > 4,627,233,987: 97613312500000000000000033658281432 15 > 4,627,233,988: 95226625000000000000000067316562865 16 > << x2 > 4,627,233,989: 90453250000000000000000134633125730 16 > << x2 > 4,627,233,990: 80906500000000000000000269266251461 17 > << x2 > 4,627,233,991: 61813000000000000000000538532502922 18 > zeros found > > One way of thinking about this is to note that the > number > 7559633198101818561553955078125, which appears on the > 2^4,627,233,965 > line, is divisible by 5^26. This means that it can be > doubled 26 times >  generating a new zero every time. However, these > zeros are eroded on > the right as the doubling proceeds. > > The program held the power of two as 154,770,693 base > 10^9 digits. The > least significant of these digits being numbered > zero. > > Here are elements 816,289 down to 816,283 of the > digits array  > showing how the 18 zeros were distributed across the > base billion > digits: > > 311542059 539263618 130000000 000000000 005385325 > 029229064 379577862 > > Of course, I realise that this result is of > negligible mathematical > interest :) > >  > Clive Tooth
Begining with the first zero the high order 3 digit pattern prior to the zeros = 125,625,125,625,125,625,125, 625,125,625,125,625,125,625,125,625,125,625,125,625,125, 625 and then the pattern changes to no pattern  325,065 and then finely 813... What's going on there?

