Re: Length of sequence of consecutive primes starting at 2
From: Robert Silverman (anonymous_at_mathforum.org)
Date: 07/20/04
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Date: Tue, 20 Jul 2004 15:12:57 +0000 (UTC)
On 20 Jul 2004, Joseph Weinberg wrote:
>rsilverman@draper.com (Bob Silverman) wrote in message news:<200407191526.i6JFQck07076@proapp.mathforum.org>...
<snip>
>"known" consecutive primes. What I mean by consecutive primes is the
>following. The sequence 2,3,5,7,11,13 is made up of consecutive
>primes while the sequence 2,3,5,11,13 is not (it's missing the prime
>number 7). Understand what I mean? And yes, I mean the largest list
>that has ACTUALLY been published.
Such a list is going to be fairly small. The largest PUBLISHED list
I know is VERY old and was published by D.N. Lehmer. It goes up
to 10 million. Modern researchers might have somewhat larger tables
stored on disk somwehere but I can't image the need to store such
a table beyond (say) 10^9 or so. Computational number theorists,
if they need primes to (say) 10^12 would generate them as needed.
>
>> (3) If you mean that you are looking for a list of ALL primes below
>> a certain bound and you want the list with the largest bound that has
>> actually been calculated, I can tell you that such a list does not
>> exist per se. Or rather, that such a list changes constantly, so you
>> are asking for a moving target. It is also one thing to calculate
>> such a list, it is another to store and publish it.
>
>OK. Can I find the largest upper bound published somewhere? It's OK
>if the bound in slightly outdated.
May I ask why you need ALL the primes? As I stated, at one time or
another all primes up to about 10^12 have been generated. I doubt if
ANYONE knows what the largest upper bound truly is. It is at least 10^12.
>
>> (4) If you want a list of primes up to some bound B, it would be
>> faster to generate it via a sieve than to read it over the internet.
>> And storage for any reasonable B would be problematic. You are
>> talking terabytes.
>
>Nope, I'm not interested in this.
Huh? This contradicts what you specifically asked for!!
You asked for the largest published table. All primes to 10^12
have been generated. It would take "about" 10^12 bytes to store the
complete table. There are roughly 10^12/log(10^12) such primes,
and all primes > ~4 x 10^9 will take more than 4 bytes apiece to
store. There are compact bit maps that can be constructed, but these
do not store the primes themselves. And even they are large.
>
>> (5) The answer to your last question is trivial. If B is the bound
>> on your list, then M41 - B is the number you are looking for.
>
>See (3) (I am looking for actual numbers, not theoretical results)
How is M41 - 10^12 a "theoretical result"??
>
>> (6) At one time or another, lists of all primes up to at least 10^12
>> have been generated. But I can't imagine anyone putting such a list
>> into permanent or even semi-permanent storage. It would be pointless.
>> Further, if at anytime someone did store such a list, extending it
>> a little bit would be trivial.
>
>Really? How would you that (I mean extend the list)? Doesn't this
>suggest that it would be trivial to find the prime following M41?
I'm not sure I understand you. You seem to be confused. The largest
table of the type you ask for has maybe 12 digits in the largest prime. M41 has over 6 million. How do you jump to the conclusion that
because I say it is easy extending the complete table from (say)
10^12 to (say) 2*10^12, that it is thereby easy to find the next
prime after M41? In the first case you are dealing with 12 digit
primes, and in the latter with 6 million digit primes.
>
>> (7) If you tell me what you REALLY want, I can and will help. But
>> your questions are too imprecise for me to discern your real desire.
>> Please. If you are going to discuss mathematics, then you need to
>> learn how to pose your questions more precisely.
>
>What I am REALLY wondering about is the following. The largest known
>primes are Mersenne primes. The largest non-Mersenne prime is much
>smaller than the largest Mersenne prime [1]. It seems to me that
>there should be many primes between these two.
May I suggest that you look up the "Prime Number Theorem"?
There will indeed be many, many primes between the two.
I am wondering whether
>there exist any efficient methods of constructing primes that would
>fall into this range.
Random primes? No.
Primes of special form? Yes.
Efficient methods exist to find primes of the form (say) k^2^n + 1.
The special form allows use of FFT based multiples and fast modular
reduction also makes the primality testing algorithm rather simple.
May I suggest that you get a book on this subject?
Hans Riesel, "Prime Numbers and Computer Methods for Factorization"
is a good book for those with just a modest math background.
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