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Topic: 18 consecutive zeros in a power of two
Replies: 18   Last Post: Oct 19, 2012 6:54 AM

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dan73

Posts: 465
From: ct
Registered: 6/14/08
Re: 18 consecutive zeros in a power of two
Posted: Oct 1, 2012 12:35 AM
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> 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 On-Line 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?



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