[seqfan] A play of twin primes?

Vladimir Shevelev shevelev at bgu.ac.il
Sat Aug 6 13:06:42 CEST 2011

Dear SeqFuns,
I would like to tell you about an astonishing experiment.
For a real  number u>1, I call a prime p a u-gap prime, if, for the next prime q>p, the interval (u*p,u*q) contains no primes. I noticed some abnormality in  the distribution of u-gap primes with respect to the values of u. Consider, say, the first 10^5 primes (in order to speak about statistics). Denote F(u)=sum_{n<=10^5: p_n is u-gap}1. Let u change with step 0.01. Then in the neighborhoods of integers we have
F(1.99)=2104, F(2.00)=2160, F(2.01)=2079;
F(2.99)=1302, F(3.00)=1539, F(3.01)=1346;
F(3.99)=  926, F(4.00)=1033, F(4.01)=  918.
On the other hand,  in the neighborhoods of half-integers we have
F(1.49)=2796, F(1.50)=2703, F(1.51)=2713;
F(2.49)=1669, F(2.50)=1580, F(2.51)=1657;
F(3.49)=1149, F(3.50)=1101, F(3.51)=1131;
F(4.49)=  802, F(4.50)=  786, F(4.51)= 809.
Thus we observe local maximums in integers and local minimums in half-integers!
If anyone to confirm these observations and to try to explain these phenomenons?
I think that it is a play of twin primes, but I have no a quite satisfactory explanation.

Vladimir Shevelev

 Shevelev Vladimir‎

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