OFFSET
1,1
COMMENTS
An old conjecture of de Polignac asserts that for any positive even integer d there are infinitely many n>0 with p_{n+1}-p_n = d.
The author has formulated the following further extension.
Conjecture: For any positive even integers d_1,...,d_k, there are infinitely many positive integers n such that p_{n+2j-1}-p_{n+2j-2} = d_j for all j=1,...,k.
For example,
p_{35209566+2j-1}-p_{35209566+2j-2} = 2 for all j = 1,...,7,
p_{19726689+2j-1}-p_{19726689+2j-2} = 6 for all j = 1,...,8,
and p_{297746+2j-1}-p_{297746+2j-2} = 2j for j = 1,2,3,4,5.
LINKS
Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
Zhi-Wei Sun, On functions taking only prime values, J. Number Theory 133(2013), 2794-2812.
Zhi-Wei Sun, On primes in arithmetic progressions, preprint, arXiv:1304.5988.
Yitang Zhang, Bounded gaps between primes, Annals of Mathematics, Pages 1121-1174 from Volume 179 (2014), Issue 3.
EXAMPLE
a(1) = 3 and a(2) = 5 since {p_3,p_4}={5,7}, {p_5,p_6}={11,13} and {p_7,p_8}={17,19} are twin prime pairs.
MATHEMATICA
n=0
Do[If[Prime[k+1]-Prime[k]==2&&Prime[k+3]-Prime[k+2]==2, n=n+1;
Print[n, " ", k]], {k, 1, 100}]
PrimePi[#]&/@Transpose[Select[Partition[Prime[Range[2500]], 4, 1], #[[4]]- #[[3]] == #[[2]]-#[[1]]==2&]][[1]] (* Harvey P. Dale, Nov 20 2013 *)
PROG
(Magma) [n: n in [1..2500] | (NthPrime(n+1)-NthPrime(n)) eq 2 and (NthPrime(n+3)-NthPrime(n+2)) eq 2]; // Vincenzo Librandi, Jun 28 2015
CROSSREFS
KEYWORD
nonn
AUTHOR
Zhi-Wei Sun, Jun 03 2013
STATUS
approved