A327378 Smallest prime p in a sequence of six consecutive primes (p,q,r,u,v,w) for which the conic section discriminant Delta < 0 for the general conic section px^2 + qy^2 + rz^2 + 2uyz + 2vxz + 2wxy = 0.

863, 1303, 2539, 2953, 3251, 3457, 4007, 4139, 4507, 5209, 5431, 5717, 7229, 7867, 7933, 9323, 9421, 11821, 12011, 12101, 12143, 12907, 12983, 13709, 13859, 14071, 14549, 15661, 16141, 16811, 17977, 18773, 18899, 19069, 19577, 20347, 20533, 21013, 21503, 21599, 22543

Offset: 1

Author

Philip Mizzi, Sep 16 2019

Keywords

• nonn

Comments

Delta = pqr + 2uvw - pu^2 - qv^2 - rw^2.

Using consecutive primes in the general conic section and computing Delta, the value is most often (~98%) > 0.

Examples

			For (p,q,r,u,v,w) = (2,3,5,7,11,13), Delta = 726 > 0. Hence, p=2 (smallest prime) is not in the sequence.
For (p,q,r,u,v,w) = (863,877,881,883,887,907), Delta = -73164 < 0. Hence, p=863 (smallest prime) is a member of the sequence.
		

Crossrefs

Cf. A000040.

Programs

Mathematica

Select[Partition[Prime@ Range[3000], 6, 1], Function[{p, q, r, u, v, w}, p q r + 2 u v w - p u^2 - q v^2 - r w^2 < 0] @@ # &][[All, 1]] (* Michael De Vlieger, Sep 30 2019 *)

PARI

lista(nn) = {forprime (p=1, nn, q = nextprime(p+1); r = nextprime(q+1); u = nextprime(r+1); v = nextprime(u+1); w = nextprime(v+1); if ((x=p*q*r + 2*u*v*w - p*u^2 - q*v^2 - r*w^2)< 0, print1(p, ", ")););} \\ Michel Marcus, Sep 18 2019

Extensions

More terms from Michel Marcus, Sep 18 2019