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

3, 5, 7, 13, 17, 37, 47, 67, 73, 97, 103, 137, 163, 167, 193, 233, 277, 293, 307, 313, 317, 347, 373, 463, 487, 503, 547, 577, 593, 607, 613, 677, 743, 787, 823, 827, 853, 883, 953, 967, 983, 997, 1087, 1117, 1123, 1237, 1367, 1423, 1447, 1523, 1543, 1613

Offset: 1

Author

Reiner Martin, Jul 12 2001

Keywords

• nonn
• easy

Comments

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

When p > 3, p^2 + 2 is never prime. - Zak Seidov, Nov 04 2013

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

Examples

			a(1) = 3 because 3^2 + 4 = 13 is prime,
a(4) = 13 because 13^2 + 4 = 173 is prime. - _Zak Seidov_, Nov 04 2013
		

Links

• Harry J. Smith, Table of n, a(n) for n = 1..1000
• Yang Ji, Several special cases of a square problem, arXiv:2105.05250 [math.GM], 2021.

Crossrefs

The corresponding primes p^2+4 are in A045637.

Subsequence of A176983.

Programs

Mathematica

Select[Prime/@Range[300], PrimeQ[ #^2+4]&]

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

Formula

a(n) = sqrt(A045637(n) - 4). - Zak Seidov, Nov 04 2013

Extensions

More terms from Larry Reeves (larryr(AT)acm.org), Jul 20 2001

Edited by Dean Hickerson, Dec 10 2002