cp's OEIS Frontend

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.

Hardy and Wright: If there are an infinite number of these primes, then there are infinitely many cubic polynomials with integer coefficients and prime discriminant. It would also resolve the open conjecture that there are infinitely many non-isomorphic elliptic curves defined over the rationals and having prime conductor.

Union of A153636 and A154291. - T. D. Noe, Jan 06 2009

Several numbers are formed in more than one way, e.g. 23, 31, 239, 499, 2687, 3299, 4027, 5323, 6079, ..., . - Robert G. Wilson v, Feb 17 2009

All terms have been checked using Sage. See A154291 for more details.

Granville: "The most desired open problem in [asymptotic sieves] is to show that 4a^3 + 27b^2 is prime for infinitely many pairs of integers a, b (this is of interest because if 4a^3 + 27b^2 is prime then it is usually the conductor of the elliptic curve y^2 = x^3 + ax + b)." - Charles R Greathouse IV, Jun 06 2013