Number of zero's in largest prime...

[Mystwalker]

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.

[paulunderwood]

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

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.

[paulunderwood]

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.

[sdbardwick]

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

[patrik]

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.

[drew]

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.

Did your teacher ask about the largest prime, or the largest prime yet discovered?

[grandpascorpion]

Uh, she clearly wrote largest-known. You even quoted it.

[drew]

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.

[jinydu]

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".

[jinydu]

For the first few Mersenne primes:

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).

[grandpascorpion]

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.