A217674 Primes p = a^2 + b^2 such that (x - a)^2 + (y - b)^2 = 2 and q - p = 4, where q = x^2 + y^2 is prime, assuming that a > b > 0 and x > y > 0.

13, 313, 613, 3613, 4513, 21013, 52813, 86113, 99013, 148513, 165313, 241513, 255613, 332113, 787513, 800113, 904513, 1073113, 1720513, 2279113, 2679613, 2940313, 3471613, 4307113, 4605613, 4789513, 5168113, 6072613, 6498013, 6716113, 7469113, 8925313

Offset: 1

Author

Thomas Ordowski, Oct 10 2012

Keywords

• nonn

Comments

The primes p and q have the smallest difference and the shortest distance on the 2-dimensional plane. Primes p = a^2 + (a-1)^2 == 13 (mod 100) and q = (a+1)^2 + (a-2)^2 == 17 (mod 100), where natural a == 3 (mod 5).

Links

• David W. Wilson, Table of n, a(n) for n = 1..1000

Crossrefs

Cf. A217503.

Programs

Mathematica

isP[p_, p1_List, p2_List] := Module[{q = Sort[Abs[p1 + p2]], s}, s = q[[1]]^2 + q[[2]]^2; q != p1 && s - p == 4 && PrimeQ[s]]; testQ[p_] := Module[{pp = PowersRepresentations[p, 2, 2][[1]]}, isP[p, pp, {-1, -1}] || isP[p, pp, {-1, 1}] || isP[p, pp, {1, -1}] || isP[p, pp, {1, 1}]]; Select[Prime[Range[200000]], Mod[#, 4] == 1 && testQ[#] &] (* T. D. Noe, Oct 19 2012 *)

Formula

A002313(n) such that A217503(n) = 2 and A082073(n-1) = 4.

Intersection of A001844 and A023200.

a(n) = A001844(m) for some m == 2 or 12 (mod 15).

a(n) == 13 (mod 300).

Extensions

Extended by T. D. Noe, Oct 19 2012