A153635 Primes of the form 4*x^3 + 27*y^2, with x positive or negative.

23, 31, 59, 139, 211, 239, 283, 419, 491, 499, 563, 643, 743, 751, 823, 1291, 1319, 1327, 1399, 1427, 1579, 1823, 1931, 2039, 2687, 2767, 3011, 3119, 3163, 3191, 3271, 3299, 3307, 3371, 3559, 3767, 3803, 3919, 4027, 4091, 4099, 4243, 4423, 4567, 4639

Offset: 1

Author

T. D. Noe, Dec 29 2008, Jan 06 2009

Keywords

• nonn

Comments

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

Examples

			1427 = 4*(-694)^3 + 27*7037^2. - _Robert G. Wilson v_, Feb 17 2009
		

References

• G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 6th Edition, Oxford Univ. Press, 2008, p. 595.

Links

• Andrew Granville, Different approaches to the distribution of primes, Milan Journal of Mathematics 78 (2009), pp. 1-25.

Crossrefs

Cf. A153636 (positive x only).

Programs

Mathematica

lst = {}; Do[ If[ z = 4x^3 + 27y^2; 0 < z < 10000 && PrimeQ@z, AppendTo[lst, z]; Print[{z, x, y}]], {y, 25000}, {x, -Floor[(27 y^2/4)^(1/3)], -Floor[(27 y^2/4)^(1/3)] + 100}]; Take[ Union@ lst, 45] (* Robert G. Wilson v, Feb 17 2009 *)

Extensions

a(23)-a(45) from Robert G. Wilson v, Feb 17 2009

Comment corrected by T. D. Noe, Jun 18 2009