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

1,1

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

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.

Formula

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

Example

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

Mathematica

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

Prog

(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

Crossrefs

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

Subsequence of A176983.

Sequence in context: A003424 A073638 A066464 * A194829 A378189 A226794

Adjacent sequences: A062321 A062322 A062323 * A062325 A062326 A062327

Keyword

nonn,easy,changed

Author

Reiner Martin, Jul 12 2001

Extensions

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

Edited by Dean Hickerson, Dec 10 2002

Status

approved