A162857 Primes of the form 4p - 1, p a prime.

7, 11, 19, 43, 67, 163, 211, 283, 331, 523, 547, 691, 787, 907, 1051, 1123, 1171, 1531, 1723, 1867, 2011, 2083, 2251, 2347, 2371, 2467, 2707, 2731, 2803, 2971, 3187, 3307, 3547, 3643, 3907, 3931, 4051, 4243, 4363, 4603, 4651, 4723, 5107, 5227, 5443, 5923

Offset: 1

Author

Daniel Tisdale, Jul 14 2009

Keywords

• nonn
• easy

Comments

If 4p - 1 is prime then n^2 + n + p = p(4p - 1) for some n = 1, 2, 3, ... [Proof. Let n + 1 = 2p, etc.]

From Alonso del Arte, Jan 14 2024: (Start)

The first six terms correspond to rings of algebraic integers of Q(sqrt(-a(n))) which are unique factorization domains.

In the ring of algebraic integers of Q(sqrt(-a(n))), the corresponding prime p = (a(n) + 1)/4 is divisible by 1/2 - sqrt(-a(n))/2 and 1/2 + sqrt(-a(n))/2, both of those being algebraic integers with minimal polynomial x^2 - x + p. For example, in Q(sqrt(-163)), we see that (1/2 - sqrt(-163)/2)(1/2 + sqrt(-163)/2) = 1/4 + 163/4 = 41, with both of the divisors having the minimal polynomial x^2 - x + 41. (End)

Crossrefs

Cf. A062737 for the corresponding primes p.

Overlaps with A003173, the Heegner numbers (last six terms of that one match the first six of this one).

Programs

Mathematica

Select[Prime[Range[1000]], PrimeQ[(# + 1)/4] &] (* Alonso del Arte, Jan 14 2024 *)

PARI

is(n)=isprime(n) && isprime(4*n-1) \\ Charles R Greathouse IV, Jun 14 2022

Scala

def isPrime(num: Int): Boolean = Math.abs(num) match {
  case 0 => false; case 1 => false; case n => (2 to Math.floor(Math.sqrt(n)).toInt) forall (p => n % p != 0)
}
((1 to 2500).map(4 *  - 1)).filter(n => isPrime(n) && isPrime((n + 1)/4)) // _Alonso del Arte, Jan 14 2024

Formula

a(n) >> n log^2 n. - Charles R Greathouse IV, Jun 14 2022

Extensions

More terms from N. J. A. Sloane, Jul 19 2009