Spacing between Prime Numbers

Date: 11/08/2005 at 09:44:25
From: Jared
Subject: Separation Between Prime Numbers

I'm aware that there is no known pattern as of yet to define the 
frequency of primes occurring; however, I'm also aware that over 
fifteen million primes have been identified and checked by reputable 
mathematics sources. 

My question is this.  Given our knowledge of primes is there a way to
determine the specific "distance" between two primes without reading
through a general list of known primes?  Even if this is not possible
(as I suspect), I'd like to know where the first place is that the
difference between two primes exceeds two thousand and how to find any
subsequent spacings of this magnitude.

The fact that no known pattern exists has it made it difficult to 
determine and locate specific spacings between primes.  The only way
I've been able to find to determine a set of primes with a specific
spacing between them has been to read through various data bases of
known primes, which has thus far been fruitless.

Date: 11/08/2005 at 14:50:08
From: Doctor Vogler
Subject: Re: Separation Between Prime Numbers

Hi Jared,

Thanks for writing to Dr. Math.  Perhaps you don't realize how much
about primes is known.  In fact, there is a simple formula that tells
us roughly how many primes will be found in a certain range, and you
can use this to say what the _average_ spacing will be.  It turns out
that the average spacing between primes around the number x will be
roughly the natural log of x.  For more details about this, see

  Prime Number Theorem
    http://mathworld.wolfram.com/PrimeNumberTheorem.html 

or search our archives or the Internet for "prime number theorem."  

On the other hand, that doesn't tell you what the individual spacings
are.  It says that they _generally_ get bigger, if slowly, but in fact
it is believed (though this has yet to be proven) that there are
infinitely many "twin primes" (which means spacings equal to two, i.e.
two primes that differ by two).  This is called the Twin Primes
Conjecture.  If it's true, then that means that the spacings get
really big and really small, and this kind of behavior tends to defy
simple formulas.

In fact, my favorite quote from

  Prime Number
    http://mathworld.wolfram.com/PrimeNumber.html 

is: 

In a 1975 lecture, D. Zagier commented "There are two facts about
the distribution of prime numbers of which I hope to convince you so
overwhelmingly that they will be permanently engraved in your hearts.
The first is that, despite their simple definition and role as the
building blocks of the natural numbers, the prime numbers grow like
weeds among the natural numbers, seeming to obey no other law than
that of chance, and nobody can predict where the next one will sprout.
The second fact is even more astonishing, for it states just the
opposite: that the prime numbers exhibit stunning regularity, that
there are laws governing their behavior, and that they obey these laws
with almost military precision" (Havil 2003, p. 171).

It is generally not hard to use factorials (or something similar) to
find a span of M consecutive integers that are all composite, but
these tend to give very large numbers.  For example, all of the
integers from

  (M+1)! + 2

to

  (M+1)! + M + 1

are composite (and this is M numbers).  But (M+1)! is a very large
number, so this will usually not give the _first_ time there is a
spacing of M, and it doesn't guarantee that the two numbers on either
end are prime; so it only shows that the spacing is _at least_ M.  To
find the first time the spacing is bigger than 2000, you have to
compute all prime numbers until you find that first spacing, and this
can take a lot of time.

But we know that the _average_ spacing won't be 2000 until you get as
high as about e^2000 (which is a huge number, with 869 digits).  Of
course, the first spacing of that size will probably show up before
the average spacing gets that big, but you should still expect that it
won't show up until you've computed more primes than all the computers
in the world today could compute in a billion years.

By comparison, the number (2002)! + 2 has 5743 digits, so this is also
a huge number, and much bigger than e^2000.

If you have any questions about this or need more help, please write
back and show me what you have been able to do, and I will try to
offer further suggestions.

- Doctor Vogler, The Math Forum
  http://mathforum.org/dr.math/