OFFSET
1,1
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.
LINKS
T. D. Noe, Sage/Python program
EXAMPLE
743 = 4*(-17977)^3 + 27*927735^2
CROSSREFS
KEYWORD
nonn
AUTHOR
T. D. Noe, Jan 06 2009, Jun 18 2009, Jun 21 2009
STATUS
approved