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 2018-01-17, 13:40 #1 venec7777   "khvicha matkava" Jan 2018 georgia 1 Posts prime number ready full justification of prime numbers their exact location distribution concept twin different from classical also new formulas for finding them and much more related to prime numbers need a site to make a presentation I hope for the help of interested persons in this field I do not know English well with a translator
 2018-01-17, 14:56 #2 lukerichards     "Luke Richards" Jan 2018 Birmingham, UK 25×32 Posts Your request does not translate well into English, but I'll give you some basic point. I will use some of the key words in your request as starting points: Distribution: Prime numbers appear to be randomly distributed. This is part of the mystery of them - it is impossible to predict where the next one is going to be. We can predict where it *might* be, but this is not certain. Twin: There are such things as 'twin primes'. Some prime numbers appear very close to other prime numbers. This is quite surprising. Not every prime has a twin. We say that a prime has a twin if the next prime (or the previous prime) is 2 more (or two less) than the last one. Examples include 3&5, 5&7, 11&13 or 18408581&18408583 Formulas: One does not exist for finding primes. There is no way to predict the next prime accurately. However, there are formulas that *might* be prime. 2p-1 is a common one. It only works when p is prime. These are called Mersenne Primes. However, there are only 50 Mersenne Primes whcih have ever been discovered, and we have checked every possible value for p up to 76,000,000 so there are lots of numbers which do not give primes. There are also formulas for *checking* such as the Lucas Lehmer test, but these are quite complicated. I hope this is helpful.
2018-01-17, 15:08   #3
science_man_88

"Forget I exist"
Jul 2009
Dumbassville

26×131 Posts

Quote:
 Originally Posted by lukerichards Distribution: Prime numbers appear to be randomly distributed. This is part of the mystery of them - it is impossible to predict where the next one is going to be. We can predict where it *might* be, but this is not certain. Twin: There are such things as 'twin primes'. Some prime numbers appear very close to other prime numbers. This is quite surprising. Not every prime has a twin. We say that a prime has a twin if the next prime (or the previous prime) is 2 more (or two less) than the last one. Examples include 3&5, 5&7, 11&13 or 18408581&18408583 Formulas: One does not exist for finding primes. There is no way to predict the next prime accurately. However, there are formulas that *might* be prime. 2p-1 is a common one. It only works when p is prime. These are called Mersenne Primes. However, there are only 50 Mersenne Primes whcih have ever been discovered, and we have checked every possible value for p up to 76,000,000 so there are lots of numbers which do not give primes. There are also formulas for *checking* such as the Lucas Lehmer test, but these are quite complicated. I hope this is helpful.
Not quite what I learned here:

There's always a prime between n and 2n
Theres a polynomial in roughly 25 variables that has all its positive values be the primes.
Lucas lehmer implementations can be complex because they use FFT, otherwise the test is fairly easy to program for.

2018-01-17, 18:04   #4
CRGreathouse

Aug 2006

3×1,993 Posts

Quote:
 Originally Posted by lukerichards Formulas: One does not exist for finding primes.
I have an unpublished (so far) list of 60+ formulas for primes, including the Jones, Sato, Wada, & Wiens polynomial that sm88 mentioned.

I think what was intended is that there are no *efficient* formulas for primes, it's faster to find them by various algorithms.

Last fiddled with by CRGreathouse on 2018-01-17 at 18:05

 2018-01-18, 15:01 #5 Dr Sardonicus     Feb 2017 Nowhere 117628 Posts If memory serves, there are "formulas" for primes in Hardy and Wright, some involving trigonometric series. They are, of course, computationally useless. There is also a formula of sorts involving Mills' Constant. The joker here is (at least if I understand correctly) that the value of the constant has to be "reverse engineered" from the sequence of primes it produces! My all-time favorite is from a Martin Gardner "Mathematical Games" column. It produces a prime for every positive integer n: 4 + (-1)n Last fiddled with by Dr Sardonicus on 2018-01-18 at 15:02 Reason: Fixing typos
 2018-01-18, 17:24 #6 CRGreathouse     Aug 2006 3·1,993 Posts Hardy & Wright has $\pi(n) = -1+\sum_{j=3}^n (j-2)! - j \left\lfloor\frac{(j-2)!}{j}\right\rfloor$ and also references some previously published formulas (Mills', I think). The result doesn't appear until the fifth edition so I think it's properly credited to Wright alone (as Hardy had been dead for 32 years when it came out).
 2018-01-19, 00:29 #7 science_man_88     "Forget I exist" Jul 2009 Dumbassville 26·131 Posts http://primes.utm.edu/glossary/ may be one good read. http://mathworld.wolfram.com may be another, forsome involved.
2018-01-23, 13:58   #8
kruoli

"Oliver"
Sep 2017
Porta Westfalica, DE

2·3·127 Posts

Quote:
 Originally Posted by CRGreathouse $\pi(n) = -1+\sum_{j=3}^n (j-2)! - j \left\lfloor\frac{(j-2)!}{j}\right\rfloor$
This formula doesn't seem to output correct values for small $$n$$. I tried to put it into a more reliable form, which (should) work $$\forall\ n \in \mathbb{N}\setminus\{0\}$$:
$\pi(n) =\sum_{i=1}^n \left\lceil\frac{(i-1)!^2}{i}\right\rceil - \left\lfloor\frac{(i-1)!^2}{i}\right\rfloor$

Maybe this is a well known formula?

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