mersenneforum.org  

Go Back   mersenneforum.org > Great Internet Mersenne Prime Search > Math > Number Theory Discussion Group

Reply
 
Thread Tools
Old 2018-01-17, 13:40   #1
venec7777
 
"khvicha matkava"
Jan 2018
georgia

1 Posts
Default 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
venec7777 is offline   Reply With Quote
Old 2018-01-17, 14:56   #2
lukerichards
 
lukerichards's Avatar
 
"Luke Richards"
Jan 2018
Birmingham, UK

283 Posts
Default

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.
lukerichards is offline   Reply With Quote
Old 2018-01-17, 15:08   #3
science_man_88
 
science_man_88's Avatar
 
"Forget I exist"
Jul 2009
Dumbassville

26·131 Posts
Default

Quote:
Originally Posted by lukerichards View Post
  • 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.
science_man_88 is offline   Reply With Quote
Old 2018-01-17, 18:04   #4
CRGreathouse
 
CRGreathouse's Avatar
 
Aug 2006

2×3×977 Posts
Default

Quote:
Originally Posted by lukerichards View Post
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
CRGreathouse is offline   Reply With Quote
Old 2018-01-18, 15:01   #5
Dr Sardonicus
 
Dr Sardonicus's Avatar
 
Feb 2017
Nowhere

2×23×71 Posts
Default

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
Dr Sardonicus is offline   Reply With Quote
Old 2018-01-18, 17:24   #6
CRGreathouse
 
CRGreathouse's Avatar
 
Aug 2006

2×3×977 Posts
Default

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).
CRGreathouse is offline   Reply With Quote
Old 2018-01-19, 00:29   #7
science_man_88
 
science_man_88's Avatar
 
"Forget I exist"
Jul 2009
Dumbassville

838410 Posts
Default

http://primes.utm.edu/glossary/ may be one good read.
http://mathworld.wolfram.com may be another, forsome involved.
science_man_88 is offline   Reply With Quote
Old 2018-01-23, 13:58   #8
kruoli
 
kruoli's Avatar
 
"Oliver"
Sep 2017
Porta Westfalica, DE

24·11 Posts
Default

Quote:
Originally Posted by CRGreathouse View Post
\[\pi(n) = -1+\sum_{j=3}^n (j-2)! - j \left\lfloor\frac{(j-2)!}{j}\right\rfloor\]
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?
kruoli is offline   Reply With Quote
Reply

Thread Tools


Similar Threads
Thread Thread Starter Forum Replies Last Post
NEW MERSENNE PRIME! LARGEST PRIME NUMBER DISCOVERED! dabaichi News 561 2013-03-29 16:55
Number of distinct prime factors of a Double Mersenne number aketilander Operazione Doppi Mersennes 1 2012-11-09 21:16
Estimating the number of prime factors a number has henryzz Math 7 2012-05-23 01:13
New prime number? inthevoid2 Information & Answers 3 2008-09-29 23:27
When do I know if the number is prime? uniqueidlondon Software 1 2003-05-17 16:57

All times are UTC. The time now is 23:13.

Thu Jul 2 23:13:23 UTC 2020 up 99 days, 20:46, 1 user, load averages: 1.15, 1.29, 1.38

Powered by vBulletin® Version 3.8.11
Copyright ©2000 - 2020, Jelsoft Enterprises Ltd.

This forum has received and complied with 0 (zero) government requests for information.

Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or any later version published by the Free Software Foundation.
A copy of the license is included in the FAQ.