Distribution of twin primes

[Meeussen Wouter (bkarnd)]

just an observation:

Mathematica_4 hates 
	(quote) n=46340 (n^2~=32bit signed integer 2.147*10^9) (endquote)

Table[{n, temp = Select[Range[n^2, (n + 1)^2], PrimeQ[#] && PrimeQ[# + 2]
&]; 
    Length[temp], temp}, {n, 46340, 46340}]

selects *negative* primes from range n^2, (n + 1)^2 :
{...... ,
2147482817, 2147482949, -2147482951, -2147482819, -2147482663, -2147482093,
\
-2147481901, -2147480971, -2147480899, -2147480299, -2147480011,
-2147479753, \
-2147479549}

Pfffrrrt,

Wouter L. J. Meeussen
Senior Scientist
N.V. Vandemoortele
L&D division
tel +32 (0)51 33 21 24




-----Original Message-----
From: Pfoertner, Hugo [mailto:Hugo.Pfoertner at muc.mtu.de]
Sent: woensdag 21 januari 2004 15:31
To: seqfan at ext.jussieu.fr
Cc: 'Ernst.Jung1 at t-online.de'; 'r.rosenthal at web.de';
'hermann.kremer at online.de'; 'nothing at abouthugo.de'
Subject: Distribution of twin primes


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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