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Old 2005-03-09, 18:37   #1
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Default Statistical Investigation of Large Digit Mersenne Primes . . . (please don't bite me)

Because I don't have the necessary skills to do a statistical analysis on patterns within large integral numbers, I thought that I might just toss this out to those of you who do.

Is there some way to determine whether at 7.9 million digit, base ten number manifests a greater degree of structure if it is a Mersenne prime than a 7.9 million digit number that was generated through some sophisticated random digit generator?

Thanks for considering this.

Simon
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Old 2005-03-09, 18:43   #2
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Best of my understanding, the digits are about as random as any random digit generator (and better than many).
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Old 2005-03-09, 18:51   #3
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Quote:
Originally Posted by 9021951
Because I don't have the necessary skills to do a statistical analysis on patterns within large integral numbers, I thought that I might just toss this out to those of you who do.

Is there some way to determine whether at 7.9 million digit, base ten number manifests a greater degree of structure if it is a Mersenne prime than a 7.9 million digit number that was generated through some sophisticated random digit generator?

Thanks for considering this.

Simon
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The question is not meaningful until you define what you mean by
"greater degree of structure". What is this???

Define what you mean by a "sophisticated random digit generator"??

Each number will either pass or not pass some prescribed set of statistical
tests.

A statistical comparison of just two instances of some object isn't very meaningful....

And the general (very vague) jist of your question seems to be aimed at
determining whether one number is somehow "more random" than another.
And this very concept is meaningless. There is no such thing as 'more random'
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Old 2005-03-09, 20:59   #4
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Thanks for your reply!

I have been gazing upon the beauty of M42, and feel that I am seeing unexpected patterns for digits that just don't seem to fit my understanding of randomness.

For example, take 553555333533, do you see its structure; its symmetry is incredible.

Believe it or not, that sequence is present, and is not contrived by me. My feeling of random just seems to be turned on its head when I see something like this.

So, does a random number generator need to have 7.9 million digits to give a similar result?

( I have stripped CR and/or LF from M42 )

Thanks!

Simon

Last fiddled with by 9021951 on 2005-03-09 at 21:07
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Old 2005-03-09, 21:20   #5
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In another thread around here, I pointed out that a large sample of truely random numbers should be random at all scales. This means that there should be about as many 1's as 7's and about as many 13's as 47's and about as many 123's as 555's....

For M42 to have the sequence that you pointed out seems non-random, but it can infact be random.
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Old 2005-03-09, 21:35   #6
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Quote:
Originally Posted by 9021951
Believe it or not, that sequence is present, and is not contrived by me. My feeling of random just seems to be turned on its head when I see something like this.
When you tell people to write down a "random" sequence of 0's and 1's, it is very likely that a run test can determine the non-randomness, because most people alternate those two numbers more often than it would be under random conditions. The chance that you have a run of five 1's (--> "11111") is 1/32 - you won't find it in most manual sequences, but eventually in truely random ones, though.

Concerning M42, there are a lot of seemingly non-random patterns in 7.9 million digits. But my guess is that in random sequences, the "frequency" of patterns is just as high.
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Old 2005-03-09, 22:27   #7
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Looking at the digits of numbers in base 10 for mathematical value kind of bothers me. Why base 10? Why not base 9? or base 11? It doesn't seem like any analytical tools based (pardon the pun) on a certain number base would provide much valuable input.

The only thing I could imagine being of any value is looking at the digits of numbers in base n for information about n

For example, the 196-algorithm is still unsolved for n=196. If someone can prove that the sequence goes on indefinitely, will the mathematical value we gain about 196 be all that important? Probably not. It will probably have much more mathematically to say about the number 10. Extending it out, if 196 in base 10 has some property, and there are other numbers n in some base b that have the same property, then maybe the set of all (n,b) will be interesting. Maybe.
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Old 2005-03-10, 01:23   #8
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I remember a similar question(IMO) that I posted on the Riesel Sieve forums about a possible prime pattern(VERY polite group, I'm simply amazed I haven't been flamed yet).

I asked them if anyone had come up with a brute force program to try and find a prime number pattern. I don't remember the precise answer, but they did say that many people have tried. I then suggested the possibility that prime numbers might possibly be a form of some sort of ultimate "anti-pattern." They very politely ignored my post, although the topic title,"Possible prime pattern" probably got the topic a fair number of hits.

Asking ignorant questions is fun if you ask the right people. Although I do intend to learn about the math this summer. I'm headed to the math forum to ask advice on what type of math courses to take to help me figure out how you determine whether equations like k*2^n-1 is prime or not.

If you have any advice, meet me there.

Edit: Oops, I'm already there.

Last fiddled with by jasong on 2005-03-10 at 01:25
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Old 2005-03-10, 16:47   #9
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Default On what I've learned based on comments . . .

Develop a thick skin made of asbestos ...

Don't reframe questions so that they are so general that people don't understand why your asking it ...

Appreciate thoughtful answers to puzzling questions ...

Anyway, I would like to think that I am following in the footsteps of the great man himself, Marin Mersenne, who even stumbled in the morass of how to confidently create prime numbers. Now, I said following, not walking by his side.

So, the base-10 comment made me think! However, without checking that what I may assert is correct, I assert that these patterns of unexpected non-randomness occur with the choice of any base; they won't be necessarily be located at precisely the same location from base to base.
(Simon's Hypothesis)

And why did I choose base-10; it is for the simple reason that it is very easy to get a base-10 printout of M42, or any of the other M xx numbers, where xx represents digits from 01 to 42, at this time. I would love to create a base-16 , base-11 or base-19 representation but I presently lack that skill, so I will only comment about something that I do have available for examination.

Now, what I also realized, was that I was rooting around in the innards of a Prime Number, and not the result of some process of multiplication. I guess, taking the metaphor a little bit further, that I reached into the body and discovered something like an appendix, and in so doing, said "What is this doing here? Is it something useful or non-functional?" So finding some structure may have some merit but of as yet, an unknown value.

So, what I need is a base-converter to check out my assertion. I also need a random-number generator that passes muster of those cynics out there, that may assert that "Heck, that ain't a random number!! It's not random enough".

Well, there it is, and now it's back to you!

Humbly submitted by,

Simon
(The Gladiator)
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Old 2005-03-10, 17:24   #10
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Quote:
Originally Posted by 9021951
Develop a thick skin made of asbestos ...

<snip>

(The Gladiator)
I suggest that everyone who has participated in this discussion run to
get a copy of Knuth's Art of Computer Programming Vol 2 and READ
the discussion in Chapter 3, especially 3.5 "What is a random sequence"

Most of the notions I have seen in this discussion are just plain WRONG.

In general, people without training in probability and statistics are very
poor judges of 'randomness'. If they see what they think is a 'pattern'
buried within a sequence, they somehow think this is an indication of
non-randomness when in fact the opposite is true.

Go READ instead of groping around in the dark.
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Old 2005-03-10, 17:38   #11
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Default Please enlighten, not denegrade

Sir, if you have a copy of this book that you mentioned, why don't you just summarize that particular section for our elucidation.

Unbeknownst to you, there are some of us that don't have ready access to that resource.

And just to make this personal, I hate your carping. Learn to be nice, or just go away, as you have recently suggested to another TROLL.

So you can now stop this here, or carry it to another level!

Simon
(The Gladiator)
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