Distribution of twin primes
Pfoertner, Hugo
Hugo.Pfoertner at muc.mtu.de
Wed Jan 21 15:31:11 CET 2004
SeqFans,
currently there is an interesting discussion in the Newsgroup
de.sci.mathematik "Primzahlen zwischen (2x-1)^2 und (2x+1)^2" (primes
between ...and...) with significant contributions from Ernst Jung, Hermann
Kremer and Rainer Rosenthal. In one of the last posts Ernst Jung mentioned
the distribution of twin prime pairs between consecutive squares and
conjectured that there is always at least one twin prime pair between
(2x-1)^2 and (2x+1)^2, x=1,2,3,... He also gave a list of intervals between
consecutive squares containing no pair of twin primes. To check his numbers
I wrote a little program and got the following results (not yet submitted):
Number of pairs of twin primes between n^2 and (n+1)^2
0 1 1 1 1 1 1 1 0 2 1 1 2 1 2 2 1 1 0 2 1 1 1 2 2 0 0 3 2 0 1 3 2 0 3 2
1 3 0 3 2 1 3 2 4 2 2 3 0 2 2 4 0 2 1 1 5 4 4 1 2 3 4 3 5 2 2 3 2 4 1 2
2 3 4 3 0 3 3 2 4 5 2 2 3 4 1 2 3 2 3 3 1 5 1 3 4 4 2 5 3 4 1 3 5 1 2 4
and
n such that there are no twin primes between n^2 and (n+1)^2.
9 19 26 27 30 34 39 49 53 77 122
The interesting thing is, that I couldn't find more terms in the latter
sequence up to n=46340 (n^2~=32bit signed integer 2.147*10^9). This suggests
the rather bold conjecture, that all intervals beyond [123^2,124^2] contain
at least one twin prime pair. Proving seems impossible. Anybody interested
in extending the checked range?
Thanks
Hugo
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