A226318 Positive integers n with p_{n+1}-p_n = 2 and p_{n+3}-p_{n+2} = 2, where p_k denotes the k-th prime.

3, 5, 26, 33, 41, 43, 81, 140, 142, 171, 176, 234, 286, 294, 313, 318, 428, 458, 473, 475, 484, 577, 579, 584, 671, 743, 772, 862, 870, 872, 891, 934, 957, 1030, 1115, 1165, 1167, 1169, 1230, 1339, 1351, 1404, 1462, 1548, 1621, 1651, 1707, 1823, 1833, 1867, 1923, 2021, 2052, 2066, 2068, 2121, 2151, 2199, 2309, 2362

Offset: 1

Author

Zhi-Wei Sun, Jun 03 2013

Keywords

• nonn

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.

Examples

			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.
		

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.

Crossrefs

Cf. A000040, A001359, A006512, A225889, A226115.

Programs

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

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 *)