A154291 Primes of the form 4x^3 + 27y^2, with x<0.

23, 31, 139, 211, 239, 419, 491, 499, 563, 643, 743, 751, 823, 1291, 1319, 1427, 1931, 2039, 2687, 2767, 3011, 3119, 3163, 3191, 3299, 3307, 3803, 3919, 4027, 4091, 4099, 4423, 4703, 4999, 5323, 5639, 5647, 6007, 6043, 6079, 6323, 6691, 6719, 6763, 7331

Offset: 1

Author

T. D. Noe, Jan 06 2009, Jun 18 2009, Jun 21 2009

Keywords

• nonn

Comments

For each prime p, the elliptic curve 27y^2 = 4x^3 + p must be solved to determine whether there is an integer solution with x positive. About 2/3 of all primes can be eliminated because p-4x^3 is never divisible by 27. The remaining primes are congruent to +-5 (mod 18). Hence this sequence is a subsequence of A129806. Half of those primes can be eliminated because even when 27 does divide p-4x^3, the quotient must equal 1 (mod 4) in order to be a square. Hence all these primes must equal 23 or 31 (mod 36). James Buddenhagen used APECS and I used Sage to examine the elliptic curves. The first difficult prime is 1831. All the elliptic curves with p = 23 or 31 (mod 36) appear to have trivial torsion and rank 0 or 2.

See the link to the Sage/Python program to see how the problem with 1831 was resolved. The first prime producing an elliptic curve of rank 4 is 19427.

Examples

			743 = 4*(-17977)^3 + 27*927735^2
		

Links

• T. D. Noe, Sage/Python program

Crossrefs

Cf. A153635, A153636