 mersenneforum.org > Math Number of zero's in largest prime...
 Register FAQ Search Today's Posts Mark Forums Read  2006-03-24, 08:50   #23
Mystwalker

Jul 2004
Potsdam, Germany

3×277 Posts Quote:
 Originally Posted by Citrix No after M43 it will be the same decimal represntation as M43 not one digit. Think about it. So since there are infinite bases, there are infinite #'s produding OO 0's. Citrix
Do we count the zeros in the decimal representation of all bases, or do we count the zeros in all bases in their own resp. representation?

If it's the first, there are definitely infinite many zeros, as there are inifinte many bases above M43, which all contain zeros. In addition the count of zeros stays constant, but I think that' not even needed here.

If it's the latter, then we have finite many, because all bases after M43 don't have any zeros, hence, there are only finite bases with finite zeros.   2006-03-24, 11:48   #24
paulunderwood

Sep 2002
Database er0rr

466610 Posts Quote:
 If it's the first, there are definitely infinite many zeros, as there are inifinte many bases above M43, which all contain zeros. In addition the count of zeros stays constant, but I think that' not even needed here.
I don't like this: What would the string 111 represent in base 11?

The ambiguous possiblilities, working base 10, are these three:

121+11+1

11+1

11+11

Quote:
 If it's the latter, then we have finite many, because all bases after M43 don't have any zeros, hence, there are only finite bases with finite zeros.
There is not much of a chance of there being a zero in M42 as a string in a base which has a large alphabet, but then there are M42 (2 through M42+1) possible bases (2 through M42+1.) giving unique strings. The first and last bases have no zeroes.   2006-03-24, 12:49   #25
paulunderwood

Sep 2002
Database er0rr

2·2,333 Posts  Quote:
 11+1
That should be 121+1 because the first two ones of 111 could be ambiguously assumed to be the elevens column, 11 of which is equal to 121.   2006-03-24, 18:56   #26
sdbardwick

Aug 2002
North San Diego Coun

11001100002 Posts Quote:
 Originally Posted by paulunderwood That should be 121+1 because the first two ones of 111 could be ambiguously assumed to be the elevens column, 11 of which is equal to 121.
For base 11, there must be a symbol for the 11th value (similar to hexadecimal), perhaps something like 0,1,2,3,4,5,6,7,8,9,A. That removes any ambiguity.

Quick examples:
111 base 11 = 121+11+1 decimal
11 base 11 = 11+1 decimal
10 base 11 = 11 decimal
100 base 11 = 121 decimal

A base 11 = 10 decimal
A0 base 11 = 110 decimal
AA base 11 = 120 decimal   2006-03-26, 16:05   #27
patrik

"Patrik Johansson"
Aug 2002
Uppsala, Sweden

52·17 Posts Quote:
 Originally Posted by paulunderwood I was asking about positive integer bases: How many zeroes in total in all bases of the largest known Mersenne prime? If you can't answer exactly then please give an educated guess.
My guess is 350 000 000 zeros.

Let p be the exponent, 30 402 457. Then the number of digits in base i is about p log 2 / log i. (Rounding to an integer towards +inf gives the exact number.)

Looking at the distribution of digits for base 10 it looks like the digits are equally likely to occur. We know that this is not true for base 2. (And if we think a bit about it we realise it is also not true for bases 4, 8, 16, 32, ....)

Assuming it were true for all bases we could estimate the number of zeros for base i to be (p log 2) / (i log i), and for all bases, (p log 2) sum_2^M(p) 1 / (i log i).

Or we could start the summing at 3 since we know our assumption is wrong for base i=2. And following Uncwilly's argument we stop the summing at base M(p) since there are no more zeros after that.

The sum can be approximated by the integral of dx / (x log x). The primitive functions are log (log x), and we get (approximate equalities):

log ( log M(p) ) - log( log 2.5 ) = log ( p log 2 ) - log ( log 2.5 ) = 16.95

After multiplying by the prefactor (p log 2) we arrive at 357 million, which I round down to 350 million since I inluded the bases 4, 8, 16 etc in the estimate.   2006-03-26, 18:36   #28
drew

Jun 2005

2·191 Posts Quote:
 Originally Posted by Heather How many Zero's are in the largest known prime number? Please if you know answer this question quickly. Thank you!
There is no largest prime. There should be no limit to the number of zeros a prime can have, either.

Last fiddled with by drew on 2006-03-26 at 18:46   2006-03-26, 18:43 #29 grandpascorpion   Jan 2005 Transdniestr 1F716 Posts Uh, she clearly wrote largest-known. You even quoted it. Last fiddled with by grandpascorpion on 2006-03-26 at 18:44   2006-03-26, 18:47   #30
drew

Jun 2005

38210 Posts Quote:
 Originally Posted by grandpascorpion Uh, she clearly wrote largest-known. You even quoted it.
Ah, yes thanks for catching that...I read the title first, which just says largest prime.   2006-03-26, 23:07   #31
jinydu

Dec 2003
Hopefully Near M48

2×3×293 Posts Quote:
 Originally Posted by patrik My guess is 350 000 000 zeros.
If that were true, it would certainly be stunningly low (by my intuition). As stated earlier in this thread, there are 913468 zeroes in the base 10 expansion. Your estimate would imply that the total number of zeroes across all bases is less than 400 times that much; which is astounding considering that there are nearly M43 bases to consider. Then again, the number of "digits" in a base does decrease as you get to larger bases.

Does anyone else want to comment on patrik's post? His reasoning appears to be right, but I haven't yet learned enough math to check it myself.

Also, it seems that it should be possible to check his argument numerically by actually computing the total number of zeroes across all bases for some of the smaller Mersenne primes. But I don't know enough programming to do that. To me, this would be a much more interesting question than the number of zeroes in base ten, since base ten seems a lot less natural (i.e. chosen by convention rather than a strong mathematical reason) than "sum across all bases".

Last fiddled with by jinydu on 2006-03-26 at 23:10   2006-03-27, 00:04 #32 jinydu   Dec 2003 Hopefully Near M48 2×3×293 Posts For the first few Mersenne primes: Code: Exponent Total number of zeroes 2 1 3 1 5 2 From there, it gets too tedious to manually ask Mathematica to write out the expansion for each individual base (in any case, Mathematica is unable to handle bases larger than 36). Last fiddled with by jinydu on 2006-03-27 at 00:05   2006-03-27, 00:24 #33 grandpascorpion   Jan 2005 Transdniestr 503 Posts Actually, you don't need to consider any bases b > sqrt(MM43) because the tens digit necessarily isn't zero and the units digit will never be zero because we're dealing with a prime number.   Thread Tools Show Printable Version Email this Page Similar Threads Thread Thread Starter Forum Replies Last Post dabaichi News 571 2020-10-26 11:02 R.D. Silverman Data 82 2013-08-14 15:58 Prime95 Miscellaneous Math 20 2008-07-29 16:58 wfgarnett3 PSearch 1 2004-06-28 20:51 McBryce Lounge 39 2003-08-12 19:35

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