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Compared to most of the people who frequent this place I suppose I am not very smart, but I do have a solid basic and somewhat intuitive understanding of mathematics.

Now prime numbers have always intrigued me, and I know for a fact that prime number must have a pattern. It is an unavoidable fact. I only wish I knew enough of higher mathematics in order to figure it out on my own and prove it once and for all. I dont though, so all I can do is state what I know and hope that someone will be kind and maybe give me more info or even one day solve the problem that has been driving my crazy for years and apparently so many other true mathematicians for long time. What is the pattern?

Here is what I know: 1- Prime numbers are a fact a simple reality. 2- Because they always appear in the same places no matter what system you use to count them i.g. base 2 base 3 base... well any base don't matter it is all the same in the end. They must have a real pattern. So What is the pattern? And more to the point, Why do so many people seem to think that they are random?

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closed as off-topic by Will Jagy, Cameron Williams, Bruno Joyal, Asaf Karagila, Zev Chonoles Feb 7 at 2:30

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Even simpler, why are heads and tails random? –  copper.hat Feb 6 at 22:50
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Randomness can mean so many things... Lack of knowledge, for example. –  JiK Feb 6 at 22:54
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There are plenty of known patterns in the primes. There are even exact formulas for the nth prime. –  Bruno Joyal Feb 6 at 22:54
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You might enjoy a book by Mark Kac, called Statistical Independence in Probability, Analysis and Number Theory. He does present the Erdos-Kac Theorem; and, you know, he was there when it happened. –  Will Jagy Feb 6 at 22:56
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As Bruno Joyal mentioned, one such "formula" can be found in this answer. It reads $$p_n = 1+\sum_{m=1}^{2^n}\left \lfloor\sqrt[n]{n}\left (\sum_{x=1}^m\left\lfloor \cos^2 \pi \frac{(x-1)!+1}{x} \right \rfloor\right)^{-1/n} \right\rfloor$$ –  Ian Mateus Feb 6 at 23:35

2 Answers 2

The primes are obviously not literally random, but viewing them as random is a surprisingly useful heuristic for coming up with plausible conjectures about the primes. So while the primes are not random, in some ways their behavior resembles random things. Terence Tao has some slides here that show how viewing primes as random seems to make accurate predictions of some of their properties. (The discussion starts at slide 10.)

If you're interested in more details, search for the "Cramér model" of primes.

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It might help if you forget about primes as such for a while, and try to decide what you mean by the word "pattern", or to put it another way, what you are prepared to accept as a pattern. To illustrate what I mean here are three possible answers to your question.

(1) The primes obviously have a pattern, namely "the pattern of primes". In other words, because they are an interesting set, that in itself means that they form a pattern. This interpretation makes the question easy but not very interesting.

(2) "Having a pattern" means "being recognisable by a DFA". In this case the primes do not form a pattern.

(3) "Having a pattern" means "being recognisable by a Turing machine". In this case the primes do have a pattern.

Presumably what you want is somewhere in between (2) and (3).

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I do not know what the DFA is and have never heard of the Turing machine. But as far as I know a pattern is something that is the result of repeating.... hmmmm you know this is harder to say than I thought it would be. A pattern is something that repeats, a traceable sequence of "events?". I am just not sure how to say it... a pattern is a pattern. –  CorBrand Feb 6 at 23:29
    
@CorBrand, yes exactly! - working out what is meant by a pattern is very difficult. When I said you should think about it, it wasn't a put-down, it was a serious suggestion. The concepts of DFA and Turing machine are somewhat technical but if you want to know about them you could start here and here. –  David Feb 6 at 23:48

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