Isolated primes
David V Feldman
Brun's famous theorem states that the sum of all twin prime
reciprocals converges. The proof makes no essential use of
the "2" in the (p,p+2) definition of twin prime, so it
works fine for pairs of primes differing by 2n for any
fixed value of n.
>From this one may conclude that for a given n, most primes p lie
in an interval [p-n, p+n] that contains no other prime.
On the other hand it's easy to construct long intervals containing
no prime at all, for example [n! + 2, n! + n].
Does any similarly simple trick explicitly produce long intervals
with a single prime in the middle? (Wilson's theorem says that
(p-1)!+1 is composite for prime p, so the standard trick above
would do it if one knew the primality of (p-1)!-1 for infinitely
many primes p, but I doubt this is known.)
David Feldman
Noam D. Elkies
David V Feldman <d...@hypatia.unh.edu> wrote:
>Does any similarly simple trick explicitly produce long intervals
>with a single prime in the middle? (Wilson's theorem says that
>(p-1)!+1 is composite for prime p, so the standard trick above
>would do it if one knew the primality of (p-1)!-1 for infinitely
>many primes p, but I doubt this is known.)
Fortunately you only need for each p the primality of
(Np+1)(p-1)!-1 for infinitely many N, and that's
a consequence of Dirichlet's theorem.
Without Dirichlet, one can get infinitely many primes
of the form N(p-1)!+1 by considering prime factors of
P(x) where P is the (p-1)!-st cyclotomic polynomial,
but I don't see an easy way of excluding the possibility
that N(p-1)!-1 is prime for all of them. So I suppose we
only get either arbitrarily isolated primes or infinitely
many twin primes...
--Noam D. Elkies <remove twin primes from e-address to reply>
Deptartment of Mathematics, Harvard University
Everett Howe
David V Feldman asked:
> Does any similarly simple trick explicitly produce long intervals
> with a single prime in the middle?
Here's a simple trick, but it may not satisfy you...
Pick your favorite n>0. Dirichlet's theorem shows that (roughly
speaking) a positive proportion of all primes are of the form (n!)k + 1.
Brun's theorem shows that for an infinite number of such primes, the
corresponding number (n!)k - 1 must be composite. And for every k for
which (n!)k + 1 is prime and (n!)k - 1 is composite, the interval
[(n!)k - n, (n!)k + n] contains just one prime.
- Everett
________________________________________________________________________
Everett Howe Center for Communications Research
(see my web page for my email address) 4320 Westerra Court
http://www.alumni.caltech.edu/~however San Diego, CA 92121
David V Feldman
I asked for an "explicit" construction of primes p within intervals
[p-n,p+n] containing no other prime. Henry Cohn pointed out to
me by e-mail that my question seems to presuppose the possibility of
explicitly constructing large primes *without any restriction*, something
no one knows how to do.
Several people provided clever solutions using Dirichlet's theorem on primes
in arithmetic progressions, but these solutions have a different character
than the well-known example I gave in my original post of a similar
construction.
So please allow me to amend my question as follows:
Can one give, by a formula, a sequence of composite numbers a_n such that
the next prime after (or before) a_n lies in an interval of diameter 2n+1
containing no other primes?
David
Imre Z. Ruzsa
> Most primes p lie in an interval [p-n, p+n] that contains no other
> prime.
>
> On the other hand it's easy to construct long intervals containing
> no prime at all, for example [n! + 2, n! + n].
>
> Does any similarly simple trick explicitly produce long intervals
> with a single prime in the middle?
This does not solve your question, but still may be of interest.
Let p_n be the n'th prime, and write d_n=p_{n+1}-p_n . So p_n is about
log n on average.
Erdos proved that there are values of n for which
min (d_n, d_{n+1}) / log n is arbitarily large, so these primes are
really very isolated.
This, and the newer results on configurations of consecutive primes I
know, are all based on some
statistical argument. We construct a block, where some numbers have
prescribed small divisors so
they are deterministically composite, and the rest are "sometimes" prime
by some improvement of
Dirichlet's theorem.
P. Erdos, Problems and results on the differences of consecutive primes,
Publ. Math. Debrecen 1 (1949), 33-37.
Imre Z. Ruzsa
ru...@renyi.hu
>