Prim numbers formula

Hurkyl

There already exist explicit formulas for the n-th prime, and a variety of algorithms for computing the n-th prime.

I should hope not -- I can factor such numbers in my head, no matter how big they are.

Hurkyl

It's already known. The primes occur precisely when we have an integer that is not a multiple of smaller integers. Aside from that obvious fact, there are bewildering array of bulk statistics known about the primes, and a great many more that are consequences of the Riemann hypothesis.

Kurret

Why is the mathematical world so focused on prime numbers, why not the general case, to find all numbers with n prime factors?

CRGreathouse

1. There are lots of formulas that generate primes; creating a new one won't guarantee that you can get a paper published, much less than make you rich.
2. Many encryption algorithms rely on the hardness of factoring large semiprimes; factoring primes is, of course, trivial.

CRGreathouse

I don't think there's any push to find (general) prime numbers -- there are too many -- but to learn how to quickly factor a number into its prime factors. This is interesting because such a factorization is unique (up to order and units), though there is still interest in factoring into coprimes and partial factorization in general.

DeaconJohn

That's a good question, Kurret (IMHO), and that's a good reply, Greathouse (IMHO).

Kurret,

A few more thoughts, adding to what Greathouse said.

One reason there is a "focus" on "prime numbers" these days is the relation with the Riemann hypothesis. For example, if we can show that the error term in the prime number theorem grows at the order of the square root of n, then we will know that the Riemann Hypothesis is true. (ref: Barry Mazur's "error term" article in this issue of the BAMS; the equivalence was shown about 100 years ago).

So, this fact reduces your question to "why is there so much interest in the Riemann hypothesis"?

Well, first of all, the RH is a very old problem.
Second, if it is true, many intereesting and beautiful facts about nature will follow. (My viewpoint is that interesting mathematics describes interesting things about nature.)
Third, everything that is known about the RH gives the impression that we a re very close to solving it.
Fourth, it's been that way for about 100 years now, so why haven't we solved it? (Riemann even wrote down an exact formula for the error term in the prime number theorem. It would seem that all we have to do is analyze that formula.)
Fifth, numerical studies of the behavour of the zeroes of the zeta function on the critical line indicate that current techniques are inadequate to solve it. The behaviour of those zeroes is just too strange and too sbutle to get a handle on. Therefore, a solution would likely result in a significant advance in our understanding of mathmatics in general.
Sixth, there is a million dollar prize for the person who resolves it.
Seventh, the name of the person who solves it is very likely to go down in history for a very long time.

On the other hand, as Greathouse pointed out, the question you ask is really of very high interest to today's mathematicians. One reason for that is IF we could find an effectgive algorithm to answer your question, THEN we would be a lot closer to finding an effective algorithm to factoring the product of two large primes in a short period of time. This would break RSA cryptography. It is highly likely that the proof would carry over to break discrete logarithm cryptography. It is quite possible that the proof would allow us to break elliptic curve cryptography.

The underlying mathematical meta-principle that you are touching on is that it is a lot easier to ask hard questions than it isw to ask "good" quesitons. A "good" question is a question that we have a hope of solving; that we feel that if we worked on it, we should be able to come up with something.

A small group of very good mathematicians are currently investigating the properties of numbers that are the product of a large number of small primes. The fact that that investigation is difficult (but doable) underlines just how hard the question that you ask probably is.

Another good example of a "hard" question is the question of whether or not there are an infinite number of Mersenne primes. For that one, there even exists a heuristic argument that there are an infinite number and it yields an asympototic estimate for how many there are!

And yet, as far as I know, nobody has any idea about how to get started on actually proving that there are an infinite number of them. Richard Guy also said this in his book "Open Problems in Number Theory" or something like that.

Oh, oh, oh, except the female star of the science fiction movie called "Proof." The "science fiction" in the movie is that she resolved the question about the finiteness of the Mersenne primes.

DJ

P.S. I welcome corrections, or even "that doesn't sound right" comments. That's how I learn.

hadi amiri 4

I have good news the discoverer of this formula is going to publish a book that contains many results like defining the set of twin prime numbers a formula about producing nth prime and many other top results .

CRGreathouse

Well twin primes are old, at least as old as Sophie Germain (early 1800s) and probably older. I don't know hold long we've had explicit formulas for generating the primes, rather than just algorithms, but surely more than a hundred years.

What, then, are these "other top results"?

DeaconJohn

You're kidding, right?

hadi amiri 4

why you don not respect others attempt
have you seen this book
why you are so judgemental

Hurkyl

If you don't want people evaluating your hype, then don't post it. What else are we supposed to do with it?

huba

There is certainly no point in making qualifying statements about a claim that has not been presented in the necessary detail.

CRGreathouse

We haven't seen the book, that's why I asked what the "other top results" were. It's hard to respect results you don't know about.

thompson03

I assume you guys are being sarcastic right?

There is no formula (function) which, given an arbitrary input will guarantee a prime output, let alone, give you the primes in order.

You can work recursively, (ie, the sieve of Eratosthanes) but its not really a formula, it just knocks out the composites one prime factor at a time.

Nor has it been proven (conclusively) that there are an infinitude of twin primes.

CRGreathouse

If you mean my post #25, I was not being sarcastic at all. If not, which posts were you referring to?

MathWorld cites Ruiz 2000:
Ruiz, S. M. "The General Term of the Prime Number Sequence and the Smarandache Prime Function." Smarandache Notions J. 11, 59-61, 2000.

[tex]p_n=1+\sum_{k=1}^{2(\lfloor n\ln n\rfloor+1)}\left[1-\left\lfloor\sum_{j=2}^k1+\lfloor s(j)\rfloor}/n\right\rfloor\right][/tex]

where
[tex]s(j)=-\frac{\sum_{s=1}^j\left(\lfloor j/s\rfloor-\lfloor(j-1)/s\rfloor\right)-2}{j}[/tex]


There are many more examples, some simpler and some more complex. Some give all primes in order, some give all distinct primes, some give the count of primes through n, etc.

No one on this thread has made that claim.

thompson03

f(x) = x ; if x is prime does too

let me rephrase,

there is no function that is not based on recursion or more fundamentally on a sieve process.

Making a function that step by step knocks out the composites is not exactly what I meant.

"My bad" on the linguistics there, sorry

CRGreathouse

I suppose then that you'd consider Wilson's Theorem to be essentially a sieve?