A162857 - OEIS

%I #19 Oct 07 2024 01:22:20

%S 7,11,19,43,67,163,211,283,331,523,547,691,787,907,1051,1123,1171,

%T 1531,1723,1867,2011,2083,2251,2347,2371,2467,2707,2731,2803,2971,

%U 3187,3307,3547,3643,3907,3931,4051,4243,4363,4603,4651,4723,5107,5227,5443,5923

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

%C 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.]

%C From _Alonso del Arte_, Jan 14 2024: (Start)

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

%C 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)

%F a(n) >> n log^2 n. - _Charles R Greathouse IV_, Jun 14 2022

%t Select[Prime[Range[1000]], PrimeQ[(# + 1)/4] &] (* _Alonso del Arte_, Jan 14 2024 *)

%o (PARI) is(n)=isprime(n) && isprime(4*n-1) \\ _Charles R Greathouse IV_, Jun 14 2022

%o (Scala) def isPrime(num: Int): Boolean = Math.abs(num) match {

%o   case 0 => false; case 1 => false; case n => (2 to Math.floor(Math.sqrt(n)).toInt) forall (p => n % p != 0)

%o }

%o ((1 to 2500).map(4 * _ - 1)).filter(n => isPrime(n) && isPrime((n + 1)/4)) // _Alonso del Arte_, Jan 14 2024

%Y Cf. A062737 for the corresponding primes p.

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

%K nonn,easy

%O 1,1

%A _Daniel Tisdale_, Jul 14 2009

%E More terms from _N. J. A. Sloane_, Jul 19 2009