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%I #25 Oct 23 2012 21:56:08
%S 13,313,613,3613,4513,21013,52813,86113,99013,148513,165313,241513,
%T 255613,332113,787513,800113,904513,1073113,1720513,2279113,2679613,
%U 2940313,3471613,4307113,4605613,4789513,5168113,6072613,6498013,6716113,7469113,8925313
%N 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.
%C 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).
%H David W. Wilson, <a href="/A217674/b217674.txt">Table of n, a(n) for n = 1..1000</a>
%F A002313(n) such that A217503(n) = 2 and A082073(n-1) = 4.
%F Intersection of A001844 and A023200.
%F a(n) = A001844(m) for some m == 2 or 12 (mod 15).
%F a(n) == 13 (mod 300).
%t 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 *)
%Y Cf. A217503.
%K nonn
%O 1,1
%A _Thomas Ordowski_, Oct 10 2012
%E Extended by _T. D. Noe_, Oct 19 2012
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