Prime numbers

[Bignose]

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.

I don't think that that is quite it. Because I've posted repeatedly that attempting to find a pattern or fast algorithm fro the primes is useful. We've learned a lot of number theory in these attempts. And even if no pattern is found, the attempts and the knowledge gained is still useful.

 

But I keep reading from Unity and Peter that there is a pattern that can speed up calculations and predict primes. But despite many months of repeated claims, neither can actually demonstrate it. So I just don't get why they insist on posting to this thread that there are patterns when that does not represent our current knowledge.

 

I go back to that powerpoint presentation that was linked some time ago. Note how that author presents it. He shows how algorithms are somewhat close to achieving the goal. That a pseudorandom algorithm has some success. But then he acknowledges that it isn't 100% successful. That our current knowledge does not have a 100% accurate algorithm.

 

This is all that I am asking for in this thread -- correct word usage. I don't want a student to come by and read this thread and be confused by these strongly worded statements about patterns in the distribution of the prime numbers and not have them backed up. I want people reading this thread to understand that until those statements can be backed up, they do not represent our current knowledge. That our current knowledge is that the primes are random.

[StringJunky]

This is all that I am asking for in this thread -- correct word usage. I don't want a student to come by and read this thread and be confused by these strongly worded statements about patterns in the distribution of the prime numbers and not have them backed up. I want people reading this thread to understand that until those statements can be backed up, they do not represent our current knowledge. That our current knowledge is that the primes are random.

You are absolutely right in your approach. I forgot this is the Maths section and the rules are specific to avoid confusion for serious learners of established principles. Apologies.

[Dekan]

Can the primes really be random? If they were, wouldn't it mean you could pick any number at random, then claim it might be prime?

 

But that obviously isn't true. For example, suppose you picked a number at random, such as 131,477,689,776,554,223,107,885. Then claimed it was prime.

Your claim could be instantly dismissed. Without calculating through all the possible divisors of that number. Because there's a rule - any number which ends in 5, is divisible by 5, and therefore is not prime.

 

This rule is plain and straightforward. It does not involve any random element. And it might show, that rules for determining prime numbers, can be devised, without resorting to "randomness".

Edited December 29, 2014 by Dekan

[Unity+]

I don't think that that is quite it. Because I've posted repeatedly that attempting to find a pattern or fast algorithm fro the primes is useful. We've learned a lot of number theory in these attempts. And even if no pattern is found, the attempts and the knowledge gained is still useful.

 

But I keep reading from Unity and Peter that there is a pattern that can speed up calculations and predict primes. But despite many months of repeated claims, neither can actually demonstrate it. So I just don't get why they insist on posting to this thread that there are patterns when that does not represent our current knowledge.

I should have clarified what I meant through out the thread. My apologies. As String Junkies has stated, I am more interested in exploring the possibility than anything. While there is no direct evidence of either side, claiming that there is no pattern in relation to our own knowledge does not mean there could be one found in the future.

[Acme]

Can the primes really be random?

Yes.

If they were, wouldn't it mean you could pick any number at random, then claim it might be prime?

No.

 

But that obviously isn't true.

What obviously isn't true?

 

For example, suppose you picked a number at random, such as 131,477,689,776,554,223,107,885. Then claimed it was prime.

This number is Prime:

92adeh4gbc4hb9dhg065ae0d20d1egc55644ee9ic3dghi6b816af76f3d5dhb76₁₉

Your claim could be instantly dismissed. Without calculating through all the possible divisors of that number. Because there's a rule - any number which ends in 5, is divisible by 5, and therefore is not prime.This rule is plain and straightforward. It does not involve any random element. And it might show, that rules for determining prime numbers, can be devised, without resorting to "randomness".

What's your plain & straightforward rule for the number I just gave?

 

I would also point out that your earlier question is a further example of your innumeracy. You confuse numeral with number when asking about ending digits base ten. The last digit base ten is just the remainder mod 10. It is also the same issue as far as 6x+1 & 6x+5 primes are concerned, which is another oft cited feature of primes. Of necessity primes must have a remainder when divided by anything, but this tells you nothing about which next number will be prime or what its remainder will be mod x.

[Unity+]

Can the primes really be random? If they were, wouldn't it mean you could pick any number at random, then claim it might be prime?

I would think probabilities would be unique to random numbers in comparison with a sequence that does have a pattern, but your statement doesn't make any logical sense, sorry to say.

[PeterJ]

 

Bignose writes - "But I keep reading from Unity and Peter that there is a pattern that can speed up calculations and predict primes. But despite many months of repeated claims, neither can actually demonstrate it. So I just don't get why they insist on posting to this thread that there are patterns when that does not represent our current knowledge."

 

Sorry, BG, I saw no request. But I do now.

 

Would you consider the falling density of the primes as we go higher up the number line to be an example? If this were instances of polio in a population we would call it a pattern.

 

I hope I've never suggested that there is a discernable pattern that would allow us to predict the next prime. I do believe that there might be a way to do this, but it would certainly not be by examing the pattern of the primes. This would be a huge red herring and dead end as far as I can tell.

 

The pattern would be found not in the primes but in the behaviour of their products. It would be the products that make the crazy music and the primes would be the missing spectral lines in the noise. This is a dynamic process that creates an evolving combination wave that has some predictable characteristics. It has a pattern.

 

This may not be what everybody here means by 'pattern', and perhaps not Unity, but it's what I mean by it.

 

Edited December 30, 2014 by PeterJ

[Waleed]

Amazing

 

Thanks for sharing

[PeterJ]

Amazing? How so?

[Prometheus]

Sorry to butt in when i know virtually nothing about prime numbers but the talk of probability piqued my interest.

 

Could it be that two concepts of probability are being conflated here?

 

In the typical frequentist sense we might run an experiment many times and observe frequencies, from which we might infer the underlying probability distribution.

 

(Personally I can't imagine the primes to be randomly distributed in this sense. That would mean if we were to run the 'experiment' again, we might get different prime numbers. Surely the primes are immutable).

 

But in terms of degrees of belief we may have reason to guess that a prime exists in a certain location on the number line and attach a probability to this belief. In this manner, and only through our ignorance, we might consider the primes random.

[PeterJ]

I find the talk of randomness and probability in relation to the primes very difficult since they are such slippery words. I think you're right that arguments start because people use them in different ways. I'd prefer to say that primes are difficult to calculate so it is often practical and convenient to treat them as being probabilistic. We're not talking decaying protons.

 

It certainly would seem odd to say that that the primes are in fact probabilistic, since this would suggest that the products of the primes are unpredictable in detail. .

[conway]

If it could be "proven" that 1 was for a certainty a prime, then would the "pattern" in Primes be clear and defined?

Edited July 10, 2015 by conway

[Acme]

If it could be "proven" that 1 was for a certainty a prime, then would the "pattern" in Primes be clear and defined?

In short, no. As you seem to imply, it's not a matter of proof at all but rather a matter of definition.

 

The History of the Primality of One: A Selection of Sources @ Journal of Integer Sequences

 

Abstract

The way mathematicians have viewed the number one (unity, the monad) has changed throughout the years. Most of the early Greeks did not view one as a number, but rather as the origin, or generator, of number. Around the time of Simon Stevin (15481620), one (and zero) were first widely viewed as numbers. This created a period of confusion about whether or not the number one was prime. In this dynamic survey, we collect a cornucopia of sources which deal directly with the question what is the smallest prime? The goal is to create a source book for studying the history of the definition of prime, especially as applied to the number one.

...

 

[conway]

Acme

 

Thanks for the information. It seems to me then that if I am a liberty to define 1 and division, then assuming 1 fits the definition of division, then it is also a prime?

[imatfaal]

Acme

 

Thanks for the information. It seems to me then that if I am a liberty to define 1 and division, then assuming 1 fits the definition of division, then it is also a prime?

 

!

Moderator Note

 

Not in the main fora you are not.

 

You have been given great liberty in your division by zero thread in the hope that you would understand how and where you were making errors. If you want to redefine things and explore then do it in speculations. For your guidance I don't believe you will be given the easy passage you got in the division by zero thread - you will need to comply with the rules regarding properly responding to counter-arguments.

 

Do not respond to this moderation within the thread