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Old 2013-02-11, 19:06   #1
Andrew
 
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Hi -

So I just had a couple of questions in general, and I'll also dump a pattern I've seen up to 2^37. (After which I need a new computer or better prog skills.)

First, I'm trying to wrap my head around why someone, or many now, have proven there are infinitely many primes, but why at the same time no one has figured out for sure if there are infinitely many Mersenne primes.

Anyone speculate? If there were a proof, do you all think it would probably boost our ability to find more Mersenne's, or would just be something we already figured, but "ok that's nice to know."

Second, I mostly just like playing around with patterns, and I'm a little more interested in that proof than finding big ones.

This one seems a bit useless so I'll throw it on the pile here.


Quote:
This is just the formula I put together to more quickly get the number I was looking for:

y + x = 2^11 and 2y - 2^11 = x -1

Basically I was looking for the "window" around 2^n such that from Y -> 2^n was the same distance as from 2^n -> 2*Y. Turns out, this can't be an integer, so I looked for the closest integers, with the latter being different from the former by 1. I tested for odd X

I wasn't looking for primes there at first, but I noticed it and did trials up to 61. Most of the 2^p's had prime X. Up to 61, all X for 2^pM were prime. Pm127 was prime too.

I noticed no non prime 2^n had prime X. I imagine this is probably explained the same way as it is explained on Wolfram that no non prime 2^n can have prime prime 2^n - 1.

These are the prime exponents I found where X is also prime:


3, 5, 7, 11, 17, 19, 23, 31, 43, 61, 79, 101,127, 167, 191, 199,313,347


also, I have't checked as closely, but it appears X and 2^n - 1 are always probably co-prime.


Probably just a stupid random pattern among a sea of prime randomness. Maybe you see something.



Last, I couldn't really test further because up to now I've been doing the bulk of my work in R, which is probably funny to a lot of you. The next best thing I know is C, but there I only have survival C, like I only have survival french.

Is the only way to go for this plain old programming languages, or are there software, and better yet, open source software that is good to use?

best

Last fiddled with by Andrew on 2013-02-11 at 19:29 Reason: add a little thing
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Old 2013-02-11, 19:38   #2
Dubslow
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Most "simple" proof of infinitudes of various forms of primes involve a product (plus or minus one) of the smaller examples of that form; that product (or the number above it) shares some properties of that form, whence you can conclude something about that number. Such a form does not occur dit products of Mersenne primes, so such a simplistic attack fails. I guess more advanced techniques also fail for whatever reason, though I can't comment on them. I can say that I would be shocked if an MP infinitude proof was practically useful.

As for arbitrary arithmetic, GMP (or MPIR) is your friend, especially regarding C/C++. Python offers "native" arbitrary precision integer arithmetic, though it uses GMP in the background (I think).
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Old 2013-02-11, 19:52   #3
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Quote:
Originally Posted by Andrew View Post
...Last, I couldn't really test further because up to now I've been doing the bulk of my work in R, which is probably funny to a lot of you. The next best thing I know is C, but there I only have survival C, like I only have survival french.

Is the only way to go for this plain old programming languages, or are there software, and better yet, open source software that is good to use?
Hi,

if you work with R, you will find that Pari/GP is reasonably easy to learn, and it makes it very easy to prototype your ideas with any wanted precision.
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Old 2013-02-11, 20:16   #4
Mr. P-1
 
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Quote:
This is just the formula I put together to more quickly get the number I was looking for:

y + x = 2^11 and 2y - 2^11 = x -1

Basically I was looking for the "window" around 2^n such that from Y -> 2^n was the same distance as from 2^n -> 2*Y. Turns out, this can't be an integer, so I looked for the closest integers, with the latter being different from the former by 1. I tested for odd X

I wasn't looking for primes there at first, but I noticed it and did trials up to 61. Most of the 2^p's had prime X. Up to 61, all X for 2^pM were prime. Pm127 was prime too.

I noticed no non prime 2^n had prime X. I imagine this is probably explained the same way as it is explained on Wolfram that no non prime 2^n can have prime prime 2^n - 1.

These are the prime exponents I found where X is also prime:


3, 5, 7, 11, 17, 19, 23, 31, 43, 61, 79, 101,127, 167, 191, 199,313,347


also, I have't checked as closely, but it appears X and 2^n - 1 are always probably co-prime.
I don't know who you're quoting, but this makes no sense whatsoever. I guess he's talking about the primality of the numbers 2p±1, 2p±3, 2p±5, 2p±7, 2p±9,..., but what he's trying to say about them I have no idea.

In my experience, people who can't express their ideas clearly rarely have clear ideas to express.

Last fiddled with by Mr. P-1 on 2013-02-11 at 20:17
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Old 2013-02-11, 20:28   #5
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Quote:
Originally Posted by Andrew View Post
First, I'm trying to wrap my head around why someone, or many now, have proven there are infinitely many primes, but why at the same time no one has figured out for sure if there are infinitely many Mersenne primes.

Anyone speculate? If there were a proof, do you all think it would probably boost our ability to find more Mersenne's, or would just be something we already figured, but "ok that's nice to know."
I find The Prime Pages to be very helpful. Listed under Mersenne it has:
Quote:
  1. Early History
  2. Perfect Numbers and a Few Theorems
  3. Table of Known Mersenne Primes
  4. The Lucas-Lehmer Test and Recent History
  5. Conjectures and Unsolved Problems
  6. See also Where is the next larger Mersenne prime? and Mersenne heuristics
  7. For remote pages on Mersennes see the Prime Links' Mersenne directory
Take a look and see what you think. I love these pages.
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Old 2013-02-11, 20:36   #6
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Quote:
Originally Posted by Mr. P-1 View Post
I don't know who you're quoting, but this makes no sense whatsoever. I guess he's talking about the primality of the numbers 2p±1, 2p±3, 2p±5, 2p±7, 2p±9,..., but what he's trying to say about them I have no idea.

In my experience, people who can't express their ideas clearly rarely have clear ideas to express.
No, sorry it was me just trying to be neat in a long post.


I started out knowing that Riemann proved that between K and 2K there is at least 1 prime number.

So I just started playing with numbers where the Mersenne Number (not just the primes) was exactly in the middle of a K and a 2K. Actually, I put 2^P in the middle, not 2^P - 1.


It's not some big idea, just pattern searching.

It turned out that [2^P - K] happened to be a prime number pretty frequently up to [P = 61]

These are the numbers where [2^P - K] are prime:

3, 5, 7, 11, 17, 19, 23, 31, 43, 61, 79, 101, 127, 167, 191, 199, 313, 347

And I didn't go to look back through all of them, but it appears at first glance the [2^P - K] and [2^N - 1] are co-prime.

Again, it's not some big idea, just something I got sidetracked with, and thought was neat.

[[***As a note, [2^N - K] and [2K - 2^N] cannot be equal and still be integers, and hence with integer K, 2^N cannot be exactly in the middle. One of [2^N - K] and [2K - 2^N] is odd and the difference is 1.***]]

It's almost like a standard deviation kinda concept, is what [2^N - K] is...

Dunno how much better I can do explaining, but it's not important. Sorry, I knew the first one wasn't a good effort of showing what I was thinking. Just lazy thinking...
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Old 2013-02-11, 21:04   #7
Dubslow
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Quote:
Originally Posted by Unregistered View Post
I started out knowing that Riemann proved that between K and 2K there is at least 1 prime number.
Whoa there buddy, you need to fact check that.
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Old 2013-02-11, 23:14   #8
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Quote:
Originally Posted by Unregistered View Post
No, sorry it was me just trying to be neat in a long post.


I started out knowing that Riemann proved that between K and 2K there is at least 1 prime number.

So I just started playing with numbers where the Mersenne Number (not just the primes) was exactly in the middle of a K and a 2K. Actually, I put 2^P in the middle, not 2^P - 1.
So K is the nearest odd number to 2/3 * 2^P.

Quote:
It's not some big idea, just pattern searching.

It turned out that [2^P - K] happened to be a prime number pretty frequently up to [P = 61]
So you're saying that the nearest odd number to 2^P/3 is frequently prime. More specifically that (2^P+1)/3 is frequently prime. These are called Wagstaff primes.

Quote:
These are the numbers where [2^P - K] are prime:

3, 5, 7, 11, 17, 19, 23, 31, 43, 61, 79, 101, 127, 167, 191, 199, 313, 347
A000978. You missed 13.

Quote:
And I didn't go to look back through all of them, but it appears at first glance the [2^P - K] and [2^N - 1] are co-prime.
What is N? Did you mean to write P? Any factor of (2^P+1)/3 must be a factor of 2^P+1 and therefore cannot be a factor of 2^P-1. So yes, they will always be coprime.
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Old 2013-02-11, 23:57   #9
Andrew
 
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Quote:
Originally Posted by Mr. P-1 View Post
So K is the nearest odd number to 2/3 * 2^P.



So you're saying that the nearest odd number to 2^P/3 is frequently prime. More specifically that (2^P+1)/3 is frequently prime. These are called Wagstaff primes.



A000978. You missed 13.



What is N? Did you mean to write P? Any factor of (2^P+1)/3 must be a factor of 2^P+1 and therefore cannot be a factor of 2^P-1. So yes, they will always be coprime.


N stood for the exponent when 2^X - 1 was not a prime exponent. And I beleive I used Pm to denote mersenne prime cases vs just mersenne numbers with a prime exponent. Just the n for a plain old Mersenne Number.


And an example:

2*43 = 86 :
86 - 64 = 22
64 - 43 = 21

so, 43<-------|<= -21|--------64---------|+22 =>|--------->86 = 43*2

similarly

2*85 = 170
170-128 = 42
128 - 85 = 43

85<-------|<= -43|--------128---------|+42 =>|--------->170 = 85*2

I'm looking at the left hand difference since it's odd.

In this case, 43 and 21

Last fiddled with by Andrew on 2013-02-11 at 23:58
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Old 2013-02-12, 03:50   #10
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Quote:
Originally Posted by Andrew View Post
N stood for the exponent when 2^X - 1 was not a prime exponent. And I beleive I used Pm to denote mersenne prime cases vs just mersenne numbers with a prime exponent. Just the n for a plain old Mersenne Number.
It is still not clear what you meant by "it appears at first glance the [2^P - K] and [2^N - 1] are co-prime". Are you saying that these appear to be coprime for any choice of N and P with P prime?

Quote:
And an example:

2*43 = 86 :
86 - 64 = 22
64 - 43 = 21

so, 43<-------|<= -21|--------64---------|+22 =>|--------->86 = 43*2

similarly

2*85 = 170
170-128 = 42
128 - 85 = 43

85<-------|<= -43|--------128---------|+42 =>|--------->170 = 85*2

I'm looking at the left hand difference since it's odd.

In this case, 43 and 21
You are still looking at the odd number closest to 2^N/3. If N is odd, then this number is (2^N+1)/3 which can be prime (as discussed above) only if N is prime. If N is even, this number is (2^N-1)/3, which is prime for N=4 and composite for all even N > 4.
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Old 2013-02-12, 04:27   #11
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Quote:
Originally Posted by Andrew View Post
First, I'm trying to wrap my head around why someone, or many now, have proven there are infinitely many primes, but why at the same time no one has figured out for sure if there are infinitely many Mersenne primes.
It basically comes down to the fact that the prime numbers are the building blocks of all numbers. Its easy to prove that you need infinitely many of them to do the job. The Mersenne numbers don't have any job to do (to the knowledge of mathematicians) that require them to be infinite.

Quote:
Anyone speculate? If there were a proof, do you all think it would probably boost our ability to find more Mersenne's, or would just be something we already figured, but "ok that's nice to know."
That is the difference between a constructive proof (one that enables us to construct, at least in principle, whatever it is we have proven exists) and a non-constructive proof (one that doesn't.) Both types of proofs exist for various theorems. What type a hypothetical proof of the infinity of Mersenne primes might be is something I don't care to speculate.

Quote:
Originally Posted by Dubslow View Post
Most "simple" proof of infinitudes of various forms of primes involve a product (plus or minus one) of the smaller examples of that form; that product (or the number above it) shares some properties of that form, whence you can conclude something about that number.
That's one way to prove the infinity of primes. There are others. Here are two proofs which use Mersenne numbers in two different ways.

Another way is to exhibit an infinite set of numbers, and show that they are pairwise co-prime. There are many such sets, for example the Fermat numbers.

Last fiddled with by Mr. P-1 on 2013-02-12 at 04:28
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