Distribution of twin primes

[f.firoozbakht at sci.ui.ac.ir]

Dear seqfans,

I have worked with Mathematica since 13 years ago.
I had faced with such a problem whenever the procedure got near to a 
ten-digit prime.
The procedure used to generated negative numbers and got returned.
This strange prime is p=2147483647.
p always makes trouble for the procedures using Mathematica.
This is common to all kind of investigations that I made on numbers.

For example you can try the following simple procedure:

In[13]:=
Table[n, {n, 2147483647 - 3, 2147483647 + 3}]

Out[13]=
{2147483644, 2147483645, 2147483646, 2147483647, -2147483648
, -2147483647, -2147483646}

I guess this is related to either:

1. A bug in Mathematica this strange prime.
2. A strange property of this prime.


I am looking forward to hearing any idea or comment in this regards.


Best wishes,

Farideh Firoozbakht 



Quoting "Meeussen Wouter (bkarnd)" <wouter.meeussen at vandemoortele.com>:

> 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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