A226461 Numbers n such that the following are six primes: 2*n^2 +- 1, 3*n^2 +- 1, 5*n^2 +- 1.

6, 2100, 20586, 669054, 745590, 6556122, 9317496, 10190796, 15648732, 18215196, 25561410, 35613990, 36710652, 38649066, 41124594, 41711874, 46576524, 48701400, 49406358, 59278296, 70038948, 74993808, 75553092, 83606418, 84182154, 88000374, 92527764, 98969052, 100691976

Offset: 1

Author

Alex Ratushnyak, Jun 08 2013

Keywords

• nonn

Comments

6 is in the sequence because the following are six primes: 71, 73, 107, 109, 179, 181.

Crossrefs

Cf. A173165.

Programs

Java

import java.math.BigInteger;
public class A226461 {
    public static void main (String[] args) {
      for (long n = 1; n < (1L << 30); n++) {
          long x = n*n*5;
          BigInteger b = BigInteger.valueOf(x+1);
          if (!b.isProbablePrime(80)) continue;
          b = BigInteger.valueOf(x-1);
          if (!b.isProbablePrime(80)) continue;
          x = n*n*2;
          b = BigInteger.valueOf(x+1);
          if (!b.isProbablePrime(80)) continue;
          b = BigInteger.valueOf(x-1);
          if (!b.isProbablePrime(80)) continue;
          x = n*n*3;
          b = BigInteger.valueOf(x+1);
          if (!b.isProbablePrime(80)) continue;
          b = BigInteger.valueOf(x-1);
          if (!b.isProbablePrime(80)) continue;
          System.out.printf("%d, ", n);
      }
    }
}

Mathematica

spQ[n_]:=AllTrue[Flatten[{2n^2+{1,-1},3n^2+{1,-1},5n^2+{1,-1}}],PrimeQ]; Select[ Range[101*10^6],spQ] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Jan 28 2015 *)