A176130 Lesser of a pair (p,p+4) of cousin primes whose arithmetic mean p+2 is a square number.

7, 79, 223, 439, 1087, 13687, 56167, 74527, 91807, 95479, 149767, 184039, 194479, 199807, 263167, 314719, 328327, 370879, 651247, 804607, 1071223, 1147039, 1238767, 1306447, 1520287, 1535119, 1718719, 2442967, 2595319, 2614687

Offset: 1

Author

Ulrich Krug (leuchtfeuer37(AT)gmx.de), Apr 09 2010

Keywords

• nonn

Comments

Necessarily p = 9 * (2*m - 1)^2 - 2.

Examples

			(7 + 11)/2 = 3^2, 1st term is prime(4) = 7.
(79 + 83)/2 = 9^2, 2nd term is prime(22) = 79.
m = 173 = prime(40): 21st term is p = 1071223 = prime(83637), p+2 = 3^4 * 5^2 * 23^2.
60th term is p = 27029599 = prime(1684797): p+2 = 3^2 * 1733^2.
		

References

• L. E. Dickson, History of the Theory of numbers, vol. 2: Diophantine Analysis, Dover Publications 2005.
• H. Pieper, Zahlen aus Primzahlen. Eine Einfuehrung in die Zahlentheorie. VEB Deutscher Verlag der Wissenschaften, 2. Aufl., 1984.
• A. Warusfel, Les nombres et leurs mystères, Edition du Seuil, Paris 1980.

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A023200, A046132, A174454.

Programs

Mathematica

Select[Range[1617]^2 - 2, And @@ PrimeQ[# + {0, 4}] &] (* Amiram Eldar, Dec 24 2019 *)

PARI

isok(n) = isprime(n) && isprime(n+4) && issquare(n+2) \\ Michel Marcus, Jul 22 2013

PARI

forstep(n=3,1e4,2,if(isprime(n^2-2)&&isprime(n^2+2),print1(n^2-2", "))) \\ Charles R Greathouse IV, Jul 23 2013

Extensions

Edited by D. S. McNeil, Nov 18 2010