Pattern to prime numbers?

[Qre:us]

Finding an exhaustive list is not exactly the same as finding infinite prime numbers, although the two problems are closely related.

The idea of semantics is quite applicable in this puzzle of a mathematical nature. As such, what one means by an ''exhaustive list'' for primes, would indicate that it is a thorough and complete list (definition of what an exhaustive list compounds to).

Prime numbers, as bowing to Euclid's Proof, are infinite. As such, if you are referring to an 'exhaustive list' of prime numbers, it would, by its very nature, infer infinite.

Thus, closely related isn't hitting the true crux of solving this problem (pattern to predicting prime numbers - not 75% of the time, not 89.97, etc).
It would have to be in either: (1) finding a measurement system that holds for the duration of...infinity (so far, base 6 only goes to a certain extent, then we see skips), (2) finding the pattern (graphical distribution, for which the R^2 value is not 100%, but, pretty darn close) but, the pattern/formula still evades us.

Unless I mistook what you meant by an "exhaustive list". (?)

 

[Qre:us]

What if there is no pattern due to the nature of division, the process through which you're puting them through? I require a thorough analysis of the nature of division (philosophically and mathematically) ASAP....

This is troublesome.... maybe i'll work on redefining division while you guys check out the more heads on way

As it could be true that the answer may very well lie at a more basic level - like division. So can we apply this to many inquiries of this world, the need for complete comprehension of the foundation before we start building the stacks. We should then go even further back, and first understand infinite if we are to target:

1st question to understanding division: dividng by 0 and by infinite. why is one null and the other zero (in most cases)? actually, how can it be that when we take an infinite series that should essentially be infinite divided by infinite, we can get a whole number?

...before we even think of how to divide them. Aristotle's take on infinite, not as actual, but potential infinite?

Then, comes the idea of limits. Oh and the philosophical take on the concept of infinite.

It could go on, just like the infinite;every puzzle of this universe can be broken down, divided into smaller and smaller infinite pieces. We must still sometimes, aim to tackle a greater whole.

Or, *mock my take-over-the-world-plan* & da whack like crack math puzzles out there....where can one start? And, why the heck would a nutso spend time wanting to?

* t-shirt reads:
(< -- I'm with freak)

 

[Qre:us]

It would have been nice to have an easy pattern. Of course it wouldn't have been something to write javascript about then.

Of course it would have! Javascripting is like, only, like the *awsomest of awsome* fun, ya-huh! And, about prime numbers! Right next to Spring Break partaying! Who, besides us, wouldn't think of that??
Are you trying to make me tingle????

 

[ygolo]

Lol?
Any number that is divisible by 1 and itself?

This would make 1 a prime, which it's not (anymore).

There are algorithms that have been developed for discoving new primes. I believe the problem is in finding an exhaustive list of primes. There is no known method of finding every one.

There is an algorithm to find all the primes. It is called Sieve of Eratosthenes. It is exactly what a naive attempt at finding all primes would be. Keep a running list of all known primes upto a value, increase the value to be checked by 1, and see if it is divisible by any of the primes in the table so-far. If it is not divisible by anything up to the sqrt of the number to be checked, then we have found another entry to the table.

The problem is that it is rather slow. The main issue is that we have to check for division by prevously found primes. The difficulty of finding factors of numbers is the key behind encryption mechanisms like RSA.

A neat way of generating large primes is by looking for Mersenne prime. Again, for those who're into cracking, many ecryption algorithms weaken themselves by only using 2 Mersenne primes for the key.

 

[Qre:us]

A neat way of generating large primes is by looking for Mersenne prime. Again, for those who're into cracking, many ecryption algorithms weaken themselves by only using 2 Mersenne primes for the key.

What are your thoughts on regularity to a prime number distribution [the other school of thought - globally looking at primes]? I.e., rather than algorithmically, which tells us how to produce primes through a steps-dependent process, one leads to the next, etc..., which doesn't really need to account for the 'big picture' (doesn't say much about the distribution).

It seems random, but, there may be a pattern. (?) Is there something to finding out this puzzle first?

 

[The_Liquid_Laser]

The idea of semantics is quite applicable in this puzzle of a mathematical nature. As such, what one means by an ''exhaustive list'' for primes, would indicate that it is a thorough and complete list (definition of what an exhaustive list compounds to).

Prime numbers, as bowing to Euclid's Proof, are infinite. As such, if you are referring to an 'exhaustive list' of prime numbers, it would, by its very nature, infer infinite.

Thus, closely related isn't hitting the true crux of solving this problem (pattern to predicting prime numbers - not 75% of the time, not 89.97, etc).
It would have to be in either: (1) finding a measurement system that holds for the duration of...infinity (so far, base 6 only goes to a certain extent, then we see skips), (2) finding the pattern (graphical distribution, for which the R^2 value is not 100%, but, pretty darn close) but, the pattern/formula still evades us.

Unless I mistook what you meant by an "exhaustive list". (?)

I meant that exhaustive implies infinite, but infinite does not imply exhaustive. Even if you had a formula which generated an infinite number of primes, that would not guarantee that you had found all of them.

 

[EcK]

Yes and I've found it!

-Is abducted by the men in black-

 

[Laurie]

I'm skimming but it's not really about a "list" because that is able to be made with a simple computer program. If you have an equation to use to solve for primes that would be what would make all prime numbers "known" even though they are infinite.

Infinite is not equal to unknown.

 

[Qre:us]

I meant that exhaustive implies infinite,but infinite does not imply exhaustive.

Infinite cannot imply exhaustive as it would negate the whole concept of infinite (boundless). Exhaustive only goes so far as explanation for what a computer by us can generate for a step-by-step sequence, for primes. Thus, it has its human limits. That doesn't mean there's 'limits' to the system of logic on which this numbers game is based. I would disagree with the word exhaustive. I am getting what you mean from a purely syntax model of logic, but, in this scenario, of primes and infinite, it is meaningless. Being or not being exhaustive. That's only relevant in terms of 'if humans wanted to generate all prime numbers'. I think the key question is: 'if humans CAN have the ability to generate all prime numbers'. I.e., the key to the lock.

Again, this is just Even if you had a formula which generated an infinite number of primes, that would not guarantee that you had found all of them.

Why would we want to find them all? Why would that logically and realistically be a goal of ours in the first place? To find all of them - literally? All prime numbers? I don't understand this premise.

It is only to find the system by which primes work. To find order in this apparant chaos that is such an anomaly in the field of such rigorous deductive logic that is mathematics.

Now, if we want to apply mathematical limits to infinite in the treatment of primes, here's a read off Wiki:
Prime number theorem - Wikipedia, the free encyclopedia

 

[BlueScreen]

I think most of my attempts at this have been looking at factor detection. If you have a number, what does it tell you about the possible factors? Can it tell you anymore than testing all primes less than its square root, or other more advanced methods? You always seem to get into the habit of testing for the presence of things though. And in doing that you are effectively doing the same thing a different way. There is a certain amount of information you need to detect no matter what approach you take. And a similar number of calculations end up being needed. Also if you can find a pattern in prime numbers, great! But I 'feel' there shouldn't be one.

So what I always wanted was a way of separating a discrete number into two discrete numbers. Or a prime into two primes. Most ways of doing this cause the same problem though. You end up testing the same number of things and needing the same amount of information. And they never scale up well enough to be any use at higher numbers.

So I've come to the conclusion it is best approached as a philosophical problem. Thousands of geniuses have attacked it mathematically, but how many scientifically. From a completely detached perspective. We are using numbers to solve numbers. Maybe we need to use something else for once.

 

[The_Liquid_Laser]

Infinite cannot imply exhaustive as it would negate the whole concept of infinite (boundless). Exhaustive only goes so far as explanation for what a computer by us can generate for a step-by-step sequence, for primes. Thus, it has its human limits. That doesn't mean there's 'limits' to the system of logic on which this numbers game is based. I would disagree with the word exhaustive. I am getting what you mean from a purely syntax model of logic, but, in this scenario, of primes and infinite, it is meaningless. Being or not being exhaustive. That's only relevant in terms of 'if humans wanted to generate all prime numbers'. I think the key question is: 'if humans CAN have the ability to generate all prime numbers'. I.e., the key to the lock.



Why would we want to find them all? Why would that logically and realistically be a goal of ours in the first place? To find all of them - literally? All prime numbers? I don't understand this premise.

It is only to find the system by which primes work. To find order in this apparant chaos that is such an anomaly in the field of such rigorous deductive logic that is mathematics.

Now, if we want to apply mathematical limits to infinite in the treatment of primes, here's a read off Wiki:
Prime number theorem - Wikipedia, the free encyclopedia

The reason why you'd want an exhaustive list is that the problem is only paritally solved otherwise. Prime numbers will always be considered something of an elusive mystery until one can show how they all can be found. Once they are all found then the mystery is totally solved.

 

[Qre:us]

Also if you can find a pattern in prime numbers, great! But I 'feel' there shouldn't be one.

Feeling aside, this may be the resolution to this 'problem'. However, only if we can answer: What would logically make us conclude that prime numbers occur at random?

We are using numbers to solve numbers. Maybe we need to use something else for once.

Like? Full understanding of the properties of a random process? And, that, by the play of statistics, within randomness, we can still find parts that may look like a pattern/order? Which doesn't necessarily *make* it so?

 

[Qre:us]

Prime numbers will always be considered something of an elusive mystery until one can show how they all can be found.

Showing how they can all be found, and actually doing it, are different matters. One gets to the base, the other exercises the propositions on which the base is built. That is why 'showing how they can all be found' is of interest to me, and why I cannot understand the relevance of actually, doing it...finding them all. Then, going backwards and saying, oh, that was the process by which it was achieved. It's a losing battle even before you start. As you will never find them all, as they are all infinite.

Once they are all found then the mystery is totally solved.

That's like saying that Pi is still unsolved because we haven't found the 'last number' of its decimal representation - well, that's cuz it never ends. But, this is not the case. We can predict how to get more and more accurate decimal representations of Pi, but, we'll never reach its 'end'. (as it is a trancendental number).

 

[The_Liquid_Laser]

Showing how they can all be found, and actually doing it, are different matters. One gets to the base, the other exercises the propositions on which the base is built. That is why 'showing how they can all be found' is of interest to me, and why I cannot understand the relevance of actually, doing it...finding them all. Then, going backwards and saying, oh, that was the process by which it was achieved. It's a losing battle even before you start. As you will never find them all, as they are all infinite.



That's like saying that Pi is still unsolved because we haven't found the 'last number' of its decimal representation - well, that's cuz it never ends. But, this is not the case. We can predict how to get more and more accurate decimal representations of Pi, but, we'll never reach its 'end'. (as it is a trancendental number).

Heh, I think you just misunderstand what I mean. For example we know what all of the natural numbers are even though we can't physically write them all down. That is what I mean for the primes.

 

[BlueScreen]

Feeling aside, this may be the resolution to this 'problem'. However, only if we can answer: What would logically make us conclude that prime numbers occur at random?



Like? Full understanding of the properties of a random process? And, that, by the play of statistics, within randomness, we can still find parts that may look like a pattern/order? Which doesn't necessarily *make* it so?

They don't occur at random. That is why I put feel, I can't logically back it up. They have a very obvious order. They are the spaces in this.

(2),4,6,8,10,12,14,16
(3),6,9,12,15
(5),10,15
(7),14
(11)
(13)

The thing is you need the multiplication tables to see the spaces. It is not random, it is just of a complexity that seems impossible to simplify. You get the improvement of removing redundant information, ie. 6,10,12,14,15 are mentioned twice, so you only need to check for 2 and 3 as factors, but you can't remove the underlying complexity, which is that you are finding spaces in a mesh made from multiplying prime numbers, and without accounting for all these possible options, you can't know it is prime. I am unaware of any method that doesn't just identify more redundant information. Eventually you hit entropy, if we haven't already. You won't simplify the problem of checking any further.

The patterns people find are there, but are normally the equivalent of checking for certain primes. They are just another check in disguise, and take the same number of calculation steps as checking for those primes would. There are many maths problems that people see as holy grails in solving it. But they are still the same problem dressed up enough that people are willing to play with numbers for years rather than break it down to the question of what are we doing and saying. You've got as little chance of solving them as the original problem. And unless philosophically the problem is suddenly seen in a way that avoids redoing it over and over in the same style of thinking, I don't believe it is solvable.

 

[ygolo]

I've been sleeping a lot.

But I should mention some functions of extreme importance to primes mentioned in my number theory class.

Totient Function -- from Wolfram MathWorld

The totient function of a prime number, p, is p-1

A more complex relation involves the Riemann zeta function and a particular Euler product.

Studying primes is a very deep subject.

 

[BlueScreen]

I've been sleeping a lot.

But I should mention some functions of extreme importance to primes mentioned in my number theory class.

Totient Function -- from Wolfram MathWorld

The totient function of a prime number, p, is p-1

A more complex relation involves the Riemann zeta function and a particular Euler product.

Studying primes is a very deep subject.

cool, I've seen them both before, but not read that deeply about them, how do they help? Like once you strip all the transforms and maths away, what is their value? What can they tell us about the problem?

Like it is all something I'm interested in, but a bit cynical about too. I'd prefer to just be the first part .

 

[ygolo]

cool, I've seen them both before, but not read that deeply about them, how do they help? Like once you strip all the transforms and maths away, what is their value? What can they tell us about the problem?

Like it is all something I'm interested in, but a bit cynical about too. I'd prefer to just be the first part .

Well, calculating the totient function is seen as an alternative to factorization. That's kind of cool I think. In a way, you are solving the same problem as factorization.

The Reimann Zeta function connects so many branches of mathematics, its har to know where to begin. It's just cool. Look at the simplicity of it.

Euler proved that Zeta function equaled a particular product of functions of primes. Which really allows us to get an idea of the density of prime numbers.

 

[BlueScreen]

Well, calculating the totient function is seen as an alternative to factorization. That's kind of cool I think. In a way, you are solving the same problem as factorization.

The Reimann Zeta function connects so many branches of mathematics, its har to know where to begin. It's just cool. Look at the simplicity of it.

Euler proved that Zeta function equaled a particular product of functions of primes. Which really allows us to get an idea of the density of prime numbers.

Yeh, Euler's part is almost intuitive. I think one time, I rederived it accidentally. It states something that is intrinsic to what a prime is. Which is what worries me about the zeta function. Could the Euler stuff more disprove the value of the zeta function's use with primes than support it. Like once you see the zeta function that way, does it really say much that can help with finding primes. And if it does, does this support the idea that it is not solvable, rather than the idea it can solve something.

 

[ygolo]

Yeh, Euler's part is almost intuitive. I think one time, I rederived it accidentally. It states something that is intrinsic to what a prime is. Which is what worries me about the zeta function. Could the Euler stuff more disprove the value of the zeta function's use with primes than support it. Like once you see the zeta function that way, does it really say much that can help with finding primes. And if it does, does this support the idea that it is not solvable, rather than the idea it can solve something.

Oh. I think the zeta function is incredibly important. You can look at the following for insight. It is really important to the distribution of primes.

On the Number of Primes Less Than a Given Magnitude - Wikipedia, the free encyclopedia

Here is a pdf of the translation:
http://www.maths.tcd.ie/pub/HistMath/People/Riemann/Zeta/EZeta.pdf