cp's OEIS Frontend

A062324 Primes p such that p^2 + 4 is also prime.

%I A062324 #38 Mar 09 2025 16:32:29
%S A062324 3,5,7,13,17,37,47,67,73,97,103,137,163,167,193,233,277,293,307,313,
%T A062324 317,347,373,463,487,503,547,577,593,607,613,677,743,787,823,827,853,
%U A062324 883,953,967,983,997,1087,1117,1123,1237,1367,1423,1447,1523,1543,1613
%N A062324 Primes p such that p^2 + 4 is also prime.
%C A062324 Equivalent to the definition: largest absolute dimension of Gaussian primes with prime coordinates. As 2 is the only even prime, the only possibility for a Gaussian prime to have prime coordinates is to be of the form +/-2 +/- I*p or +/-p +/-2*I with p^2+4 a prime, i.e., p is a member of this sequence. - _Olivier Gérard_, Aug 17 2013
%C A062324 When p > 3, p^2 + 2 is never prime. - _Zak Seidov_, Nov 04 2013
%C A062324 For p > 5 and q = p^2 + 4, the following congruences apply: q == 3 (mod 10) and q == 5 (mod 12). - _Joseph Wheat_, Feb 28 2025
%H A062324 Harry J. Smith, <a href="/A062324/b062324.txt">Table of n, a(n) for n = 1..1000</a>
%H A062324 Yang Ji, <a href="https://arxiv.org/abs/2105.05250">Several special cases of a square problem</a>, arXiv:2105.05250 [math.GM], 2021.
%F A062324 a(n) = sqrt(A045637(n) - 4). - _Zak Seidov_, Nov 04 2013
%e A062324 a(1) = 3 because 3^2 + 4 = 13 is prime,
%e A062324 a(4) = 13 because 13^2 + 4 = 173 is prime. - _Zak Seidov_, Nov 04 2013
%t A062324 Select[Prime/@Range[300], PrimeQ[ #^2+4]&]
%o A062324 (PARI) { n=0; forprime (p=2, 5*10^5, if (isprime(p^2 + 4), write("b062324.txt", n++, " ", p); if (n==1000, break)) ) } \\ _Harry J. Smith_, Aug 04 2009
%Y A062324 The corresponding primes p^2+4 are in A045637.
%Y A062324 Subsequence of A176983.
%K A062324 nonn,easy
%O A062324 1,1
%A A062324 _Reiner Martin_, Jul 12 2001
%E A062324 More terms from Larry Reeves (larryr(AT)acm.org), Jul 20 2001
%E A062324 Edited by _Dean Hickerson_, Dec 10 2002