18921Re: [PrimeNumbers] Known prime gaps
- May 7, 2007All,
Years ago I plotted a frequency distribution of prime gaps from 2 to some small limit and the curve always looked similar to the curve for black body radiation. Has anyone done this for limits large or small? Are there theoretical reasons to account for this?
Sent: Sun, 6 May 2007 5:14 PM
Subject: [PrimeNumbers] Known prime gaps
Polignac's conjecture says all even prime gap sizes occur infinitely many
times. So far the only known way to prove existence of a gap size is to find
Thomas R. Nicely maintains tables of first known occurrence prime gaps at
For each gap size the smallest known consecutive primes or prp's with that
gap are listed.
Torbjörn Alm has searched first known occurrence gaps for a long time with a
sieve by me, using modular equations to ensure unusually many small factors
in wanted gaps. Small prp tests are made by the GMP library, and large by
There is now a proven occurrence of all 10000 even gaps up to 20000.
Marcel Martin's Primo proved the large majority of the 20000 gap ends.
In addition, there is now either a proven or prp occurrence of all even gaps
up to 30000, and currently of 30046 even gaps in total (and 1 odd!).
Torbjörn found the listed occurrence of 21274 of them. Others had previously
found larger primes for some of the gap sizes. It is not recorded who was
the first to find an occurrence of a gap.
The Top-20 Prime Gaps at
http://hjem.get2net.dk/jka/math/primegaps/gaps20.htm lists the best gaps in
The merit of the gap from p1 to p2 is defined as (p2-p1)/log p1, where log
p1 is the average gap size in that vicinity.
This year Torbjörn has found the 3 largest known gaps with merit above 20.
The best is a gap of 114554 between 2227-digit primes. The merit is 22.34.
Jens Kruse Andersen
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