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". (?)
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Thread: Pattern to prime numbers?

02042009, 12:54 AM #21

02042009, 01:11 AM #22
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?
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 takeovertheworldplan* & da whack like crack math puzzles out there....where can one start? And, why the heck would a nutso spend time wanting to?
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02042009, 01:29 AM #23

02042009, 01:45 AM #24
This would make 1 a prime, which it's not (anymore).
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 sofar. 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.
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02042009, 02:20 AM #25
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 stepsdependent 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?

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02042009, 11:27 AM #27
Yes and I've found it!
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02042009, 11:58 AM #28
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.

02042009, 10:53 PM #29
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 stepbystep 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.
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

02052009, 03:46 AM #30
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.Freude, schöner Götterfunken Tochter aus Elysium, Wir betreten feuertrunken, Himmlische, dein Heiligtum! Deine Zauber binden wieder Was die Mode streng geteilt; Alle Menschen werden Brüder, Wo dein sanfter Flügel weilt.
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