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Prime numbers


sunshaker
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I don't know why you are getting defensive, I was simply stating a point. :(

 

EDIT: Couldn't we bring the Sieve of Erathosthenes into question in regards to its state of randomness?(Please don't attack me for this, it's just a question).

Ya. Your point was that 'seemingly random constructs have some pattern behind them'. And I really object to stating that when there is no evidence of it. I keep asking you guys to back these statements up. If there is a pattern, present it.

 

I'm not going to support any assertions about patterns until they can be supported. Mathematicians have good reason to believe NP v. P, even though it isn't proven. But I have not seen any real mathematician say they believe in a pattern in the primes without evidence.

 

Once again, this does not mean that there is no value in looking for one. But, apart from a few of you guys, I don't see anyone else insisting on there being a pattern, especially when you guys can't back these statements up.

 

That's what I'm saying. That's pretty much all that I've said in this thread. That you guys can believe whatever you want, but unless you're prepared to back patterns up with something definitive, the current state of our knowledge is that primes are distributed randomly. Saying anything else good against the very best of our current knowledge.

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Ya. Your point was that 'seemingly random constructs have some pattern behind them'. And I really object to stating that when there is no evidence of it. I keep asking you guys to back these statements up. If there is a pattern, present it.

Actually, my point was to state that there are many assumptions, whether orthodox or not, that we make, such as assuming that P are NP problems are not the same, because the contrary is less likely to be the truth. However, since there is no evidence, there is no definitive answer to give. This applies to primes as well. Whether a majority of users here belief or disbelief that there is a pattern in primes does not reflect the true nature of problems or the majority opinion on the matter.

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Just thinking about patterns in prime numbers - obviously they all have a terminal digit which is 1, 3, 7, or 9.

 

Have any studies been made, as to whether these terminal digits are evenly distributed among the prime numbers. Or do the prime numbers favour certain digits, eg, are there are more primes ending with a "1", than a "9"?

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Well, the primes are bound to make local 'patterns' by serendipity, just like the digits of pi, but they are meaningless. It's only the predictable patterns that are interesting. There are predictable aspects of prime behaviour and very clear patterns. This would be why Fourier analysis is relevant. No?

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However, since there is no evidence, there is no definitive answer to give. This applies to primes as well.

I don't think it is fair to say that there is 'no evidence'. We have all the evidence of the set of the currently known primes. And to date, no pattern or algorithm has been found to be able to predict this full set of primes.

 

This is why I harp so strongly on the state of our current knowledge. We have a lot of knowledge. And, to date, that knowledge is a great deal of evidence that the primes are random. Again, this could be shown wrong some day, but you can't describe our current knowledge as 'no evidence', because there has been a great deal of work and knowledge created to date.

 

No amount of wordsmithing or trying to subtlety trying to twist our current knowledge is going to get me to agree with you on this. What would get to agree is demonstration of a pattern. Something I've asked for for the 6+ months this thread has been open.

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I don't think it is fair to say that there is 'no evidence'. We have all the evidence of the set of the currently known primes. And to date, no pattern or algorithm has been found to be able to predict this full set of primes.

 

This is why I harp so strongly on the state of our current knowledge. We have a lot of knowledge. And, to date, that knowledge is a great deal of evidence that the primes are random. Again, this could be shown wrong some day, but you can't describe our current knowledge as 'no evidence', because there has been a great deal of work and knowledge created to date.

 

No amount of wordsmithing or trying to subtlety trying to twist our current knowledge is going to get me to agree with you on this. What would get to agree is demonstration of a pattern. Something I've asked for for the 6+ months this thread has been open.

There is a difference between circumstantial evidence and direct evidence. Direct evidence is the evidence required to make a definitive answer.

 

Also, the Sieve of Eratosthenes can be seen as a pattern, a complex pattern, but a pattern nonetheless:

 

new%20sieve.gif

 

 

 

A pattern, apart from the term's use to mean "Template",[a] is a discernible regularity in the world or in a manmade design. As such, the elements of a pattern repeat in a predictable manner.

The prediction here is that the prime numbers will not occur at divisions at 2, 3, 5, or 7. Therefore, the pattern is discernible, very complex, but is visible.

Edited by Unity+
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There is a difference between circumstantial evidence and direct evidence. Direct evidence is the evidence required to make a definitive answer.

 

Also, the Sieve of Eratosthenes can be seen as a pattern, a complex pattern, but a pattern nonetheless:

 

new%20sieve.gif

 

The prediction here is that the prime numbers will not occur at divisions at 2, 3, 5, or 7. Therefore, the pattern is discernible, very complex, but is visible.

Very complex=random. Change the number of columns and you get an entirely different pattern. No such pattern is predictive of where the next prime lies regardless of indications of where it does not lie. See the earlier article I posted on prime gaps.
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Very complex=random. Change the number of columns and you get an entirely different pattern. No such pattern is predictive of where the next prime lies regardless of indications of where it does not lie. See the earlier article I posted on prime gaps.

Very complex does not equal random, which is why they are in two different categories. :huh:

 

 

No such pattern is predictive of where the next prime lies regardless of indications of where it does not lie.

A majority of patterns rely on deductive reasoning, where it does not lie is where it will lie. For example, where there is no even number there is an odd number and where there is no odd number is an even number because a pattern does exist that allows us to make this determination.

 

I am also aware of the prime gap, I am also aware of someone proving the limit in the prime gap. I believe it is 70,000,000

 

 

From this prime-gap limit, we can discern that there will be no consecutive prime that will be larger than this limit. Therefore, randomness cannot apply to prime numbers because in order for randomness to apply to prime numbers, it must have all possible distances, or gaps, between the prime numbers.

 

EDIT: Basically, this video gives an idea about the proof, which states that on the number line, P1 - P2 < 70,000,000.

 

My bad, misinterpretation.

Edited by Unity+
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Very complex does not equal random, which is why they are in two different categories. :huh:

Says you. This is no more than your constant arguing about a definition of random. That you don't understand changes nothing about primes.

A majority of patterns rely on deductive reasoning, where it does not lie is where it will lie. For example, where there is no even number there is an odd number and where there is no odd number is an even number because a pattern does exist that allows us to make this determination.

Word salad. Your table tells you nothing about a table of 236586136850133651357137673471513 columns and since you cannot even visually take in such a table you can't use it deductively or otherwise to tell you jack squat about where primes will appear in/on it.

 

I am also aware of the prime gap, I am also aware of someone proving the limit in the prime gap. I believe it is 70,000,000

 

From this prime-gap limit, we can discern that there will be no consecutive prime that will be larger than this limit. Therefore, randomness cannot apply to prime numbers because in order for randomness to apply to prime numbers, it must have all possible distances, or gaps, between the prime numbers.

No. That bit is about the minimum gap between primes and is focused on proving the twin prime conjecture. The piece I posted on is about the maximum gap between primes.
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Says you. This is no more than your constant arguing about a definition of random. That you don't understand changes nothing about primes.

I have no comment on your statement since it is not productive in the discussion.

 

 

Word salad. Your table tells you nothing about a table of 236586136850133651357137673471513 columns and since you cannot even visually take in such a table you can't use it deductively or otherwise to tell you jack squat about where primes will appear in/on it.

The point of a pattern is it doesn't require a table to determine what is prime and what is not. It may require intensive algorithms, but is a deductive mechanism.

 

The primes are not randomly distributed. They are completely deterministic in the sense that the

nth prime can be found via sieving. We speak loosely of the probability that a given number n is prime (p (nP)1/logn) based on the prime number theorem but this does not change matters and is largely a convenience.

Some confusion is maybe due to the use of probabilistic methods to prove interesting things about primes and because once we put the sieve aside the primes are pretty inscrutable. They seem random in the sense that we cannot predict their appearance in some formulaic way.

On the other hand the primes have properties associated more or less directly with random numbers. It has been shown that the form of the "explicit formulas" (such as that of von Mangoldt) obeyed by zeros of the ζfunction imply what is known as the GUE hypothesis: roughly speaking the zeros of the ζ function are spaced in a non-random way. The eigenvalues of certain types of random matrices share this property with the zeros. There is a proof of this.1

So it can be said that the primes are a deterministic sequence that via the ζ function share a salient feature with putatively random sequences.

In response to the particular question, "random" here is the "random" of random matrix theory. The paper trail is pretty clear from the work below and it's not a subject that fits into an answer box.

1 Rudnick and Sarnak, Zeros of Principal L-Functions and Random Matrix Theory, Duke Math. J., vol. 81 no. 2 (1996).

http://math.stackexchange.com/questions/421353/are-primes-randomly-distributed

Edited by Unity+
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I have no comment on your statement since it is not productive in the discussion.

The point of a pattern is it doesn't require a table to determine what is prime and what is not. It may require intensive algorithms, but is a deductive mechanism.

More word salad. Your constant arguing is what is adding nothing to the discussion of primes. Perhaps it's just your tender age, but whatever the reason it is worthless here. Now be a good boy and go back and actually read the article I posted as well as the paper that I linked to. They are in post #89
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More word salad. Your constant arguing is what is adding nothing to the discussion of primes. Perhaps it's just your tender age, but whatever the reason it is worthless here. Now be a good boy and go back and actually read the article I posted as well as the paper that I linked to. They are in post #89

Yes, I read it many times before and failed to address it because besides this information:

 

 

 

Gaps of size (log X)2 are what would occur if the prime numbers behaved like a collection of random numbers

Which has the only relevancy of revealing the uncertainty of prime numbers, the article only describes the uncertainties of prime numbers.

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Yes, I read it many times before and failed to address it because besides this information:

 

Which has the only relevancy of revealing the uncertainty of prime numbers, the article only describes the uncertainties of prime numbers.

You're starting to babble...again. The topic of this thread is the properties of primes which covers far more than their distribution. You and the other whiners that keep carrying on about whether or not that distribution should be called random add nothing to the topic. You et al don't agree with others of us; we get it. Now [please] just shut up about it and move on with the topic.
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There is a difference between circumstantial evidence and direct evidence. Direct evidence is the evidence required to make a definitive answer.

Well, I guess this is progress as your quote before was there was no evidence. Now it is apparently circumstantial. Not sure exactly what that means in a mathematical sense. But, at least your admission that there is more than 'no' evidence, is a step in the right direction.

 

Also, the Sieve of Eratosthenes can be seen as a pattern, a complex pattern, but a pattern nonetheless:

The Sieve just uses the tautology that the sequence of prime numbers are each prime. Again, what this thread has been wholly about is finding a quicker algorithm or pattern or function to predict all the primes without directly implementing the definition of primes. This thread is not about finding the superficial patterns like "prime numbers will not occur at divisions at 2, 3, 5, or 7." We all know that, and understand that. The question at hand is without doing this directly, can you build a sequence that contains every prime number?

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The Sieve just uses the tautology that the sequence of prime numbers are each prime. Again, what this thread has been wholly about is finding a quicker algorithm or pattern or function to predict all the primes without directly implementing the definition of primes. This thread is not about finding the superficial patterns like "prime numbers will not occur at divisions at 2, 3, 5, or 7." We all know that, and understand that. The question at hand is without doing this directly, can you build a sequence that contains every prime number?

Well, if you want a formula for all prime numbers I can give you none. It was not what I claimed in the beginning.

 

 

Well, I guess this is progress as your quote before was there was no evidence. Now it is apparently circumstantial. Not sure exactly what that means in a mathematical sense. But, at least your admission that there is more than 'no' evidence, is a step in the right direction.

It was assumed that evidence/proof is always direct within Mathematics. Circumstantial evidence is irrelevant unless one is investigating the proof itself. On a further note, there is evidence, by your definition, of both sides of the argument. I don't see your point though.

Edited by Unity+
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I think the difference between Unity's view and the others is that the latter aren't interested and accept the expert consensus thus far, but he's interested and exploring it.

Yeah; I revisited the topic and posted the new finding in post #89 because I'm not interested. :rolleyes: Heaven forbid that anyone actually discuss it.
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No idea, but Unity is interested even if he doesn't manage it. Even if one doesn't achieve ones goal exploring we always learn something new about the landscape we've just explored.

So you have nothing to discuss that is on topic either. Fair enough.
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