A327592 Smallest prime (p) of six consecutive primes (p,q,r,u,v,w) for which the conic section discriminant (Delta) is a perfect square.

397, 68219, 87881, 316531, 430487, 440653, 639701, 691813, 732497, 982981, 1145773, 1226683, 1288337, 1291223, 1537751, 1563943, 1756663, 1913803, 2043397, 2134589, 2143391, 2317097, 2366789, 2528833, 3047311, 3107597, 3261523, 3678869, 3884389, 4143397

Offset: 1

Author

Philip Mizzi, Sep 18 2019

Keywords

• nonn

Comments

Delta = pqr + 2uvw - pu^2 - qv^2 - rw^2 for the general conic section px^2 + qy^2 + rz^2 + 2uyz + 2vxz + 2wxy = 0.

Perfect squares of this form are quite rare, representing approximately 0.0048% of possible Delta values using consecutive prime number coefficients. (First 4 million primes tested.)

Examples

			48 = sqrt(2304) = pqr + 2uvw - pu^2 - qv^2 - rw^2 for (p,q,r,u,v,w) = (440653,440669,440677,440681,440683,440711), which are consecutive primes. Hence, 440653 is a member of the sequence.
		

Links

• Giovanni Resta, Table of n, a(n) for n = 1..10000 (first 192 terms from Philip Mizzi)

Crossrefs

Cf. A000040.

Programs

Mathematica

f[{p_, q_, r_, u_, v_, w_}] := p q r + 2 u v w - p u^2 - q v^2 - r w^2; First /@ Select[Partition[ Prime@ Range@ 300000, 6, 1], IntegerQ@ Sqrt@ f@ # &] (* Giovanni Resta, Sep 30 2019 *)

PARI

chk(nn) = {forprime (p=1, nn, my(q = nextprime(p+1), r = nextprime(q+1), u = nextprime(r+1), v = nextprime(u+1), w = nextprime(v+1)); if (issquare(p*q*r + 2*u*v*w - p*u^2 - q*v^2 - r*w^2), print1(p, ", ")););} \\ Michel Marcus, Sep 30 2019