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Old 2006-11-17, 20:58   #1
Terence Schraut
 
Oct 2006

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Default Prime gaps

I have been searching through a number of websites about primes and have found no references to any theory predicting gaps between primes. I have a method which predicts certain series of numbers will all be composite, indicating a gap of at least a certain size.

Does anyone know about methods I haven't found? It would be helpful not to waste everyone's time with if I am duplicating someone else's work.
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Old 2006-11-17, 21:12   #2
paulunderwood
 
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See: http://mathworld.wolfram.com/PrimeGaps.html

Since you have not stated your method, it is difficult to say anything about methods you have not found

Last fiddled with by paulunderwood on 2006-11-17 at 21:15
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Old 2006-11-21, 17:46   #3
Terence Schraut
 
Oct 2006

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My method comes from studying the digits multi-digit primes end with in various number bases, with particular attention to the spacing patterns for permissible ending digits in various bases. For example 1, 3, 7, 9 in base ten as opposed to 1, 7, 11, 13, 17, 19, 23, 29 in base thirty.

Currently I can generate sequences of consecutive prime numbers of any specified length. However, sometimes they are the entire prime gap and sometimes these sequences are only the major portion of a prime gap. I do have some ideas of how to over come this.
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Old 2006-11-21, 18:23   #4
grandpascorpion
 
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So for base n, find all the d such that gcd(d,n)=1

That alone won't get you anywhere.
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Old 2006-11-21, 18:27   #5
brunoparga
 
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I don't know much math, but I guess that if he indeed had something on this, it'd be sorta the Big Holy Shining Grail of Mathematics.
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Old 2006-11-21, 20:05   #6
Ethan Hansen
 
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... have found no references to any theory predicting gaps between primes.

A good reference is HG Diamond's Elementary methods in the study of the distribution of prime numbers, Bull. of American Math. Soc #7 (1982). If you have a MathSciNet account, you should be able to retrieve it, otherwise Google can probably unearth a copy. More classic ones include Erdos and AE Ingham's Distribution of Prime Numbers (from the 20's or early 30's if my fuzzy memory is any good).

An interesting, newer reference is by I. Pritsker, treating the distributions of primes as a weighted capacity problem. An accessible online version is here
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Old 2006-11-22, 02:34   #7
Jens K Andersen
 
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Quote:
Originally Posted by Terence Schraut View Post
Currently I can generate sequences of consecutive prime numbers of any specified length. However, sometimes they are the entire prime gap and sometimes these sequences are only the major portion of a prime gap.
It sounds like you meant to write consecutive composite numbers.
If so, there are well-known methods for that. For example, n!+2 to n!+n is composite for any n>1, since n!+k with k<=n is divisible k.
Using the primorial n# = product of all prime numbers <=n, there is the similar n#+2 to n#+n (if n#+/-1 happens to be composite, then n#-n to n#+n).
n# < n! (for n>3), but if you want large gaps between relatively small primes then there are even better methods. They are based on carefully choosing which numbers in some interval of specified length should be divisible by each small prime. Then the chinese remainder theorem can be used to compute the location of such intervals. Using n! or n# are special cases of these methods. Better variants were used to find most of the largest known gaps at The Top-20 Prime Gaps.
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Old 2006-11-23, 13:46   #8
Terence Schraut
 
Oct 2006

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Dear grandpascorpion
If you look at the permissible digits for base thirty again, you may notice two regularly occuring small groups of non-prime numbers, one bounded by ((30*n)+1) and ((30*n)+7), the second bounded by ((30*n)-7) and ((30*n)-1). Knowing why these small groups occur it is easier to find larger groups of non-prime numbers.
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Old 2006-11-23, 16:05   #9
grandpascorpion
 
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Dear Terence,

"Why" was already explained in my last post. Did you understand it?

It's not easier to find anything worth finding. Trust me. You will save yourself some time.
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Old 2006-12-02, 20:50   #10
M_Hammer_Kruse
 

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The standard method to show there are prime gaps of arbitrary length is:

Look at th sequence n!+2, n!+3, .., n!+n. These numbers will be divisible by 2, 3, ..,n. Thus you found a prime gap of n-1 natural numbers.

regards, mike
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Old 2020-09-01, 23:49   #11
Bobby Jacobs
 
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Cool!
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