A166246 Primes representable as the sum of two rational cubes.

2, 7, 13, 17, 19, 31, 37, 43, 53, 61, 67, 71, 79, 89, 97, 103, 107, 127, 139, 151, 157, 163, 179, 193, 197, 211, 223, 229, 233, 241, 251, 269, 271, 277, 283, 313, 331, 337, 349, 359, 367, 373, 379, 397, 409, 421, 431, 433, 439, 449, 457, 463, 467, 499, 503, 521

Offset: 1

Author

Max Alekseyev, Oct 10 2009

Keywords

• nonn

Comments

The prime elements of A159843, i.e., the intersection of A159843 and A000040.

Also, the prime elements of A020898.

References

• H. Cohen, Number Theory. I, Tools and Diophantine Equations, Springer-Verlag, 2007, p. 378.

Links

• Fernando Rodriguez Villegas, Don Zagier, Which primes are sums of two cubes?, CMS Conference Proceedings 15 (1995), pp. 295-306.

Crossrefs

Cf. A166243, A166244, A159843.

Programs

Mathematica

(* To speed up computation, a few terms are pre-computed *) nmax = 521; xmax = 360; preComputed = {127, 271, 379}; solQ[p_] := Do[ If[ IntegerQ[z = Root[-x^3 - y^3 + p*#^3 & , 1]], Print[p, {x, y, z}]; Return[True]], {x, 2, xmax}, {y, x, xmax}]; A166246 = Union[ preComputed, Select[ Prime[ Range[ PrimePi[nmax]]], Mod[#, 9] == 4 || Mod[#, 9] == 7 || Mod[#, 9] == 8 || solQ[#] === True & ]](* Jean-François Alcover, Apr 04 2012, after given formula *)

Formula

Under the Birch and Swinnerton-Dyer conjecture, these primes consist of:

(i) p = 2;

(ii) p == 4, 7, or 8 (mod 9);

(iii) p == 1 (mod 9) and p divides A206309(p-1), i.e., Villegas-Zagier polynomial A166243((p-1)/3) evaluated at x=0.