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

1,1

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

Table of n, a(n) for n=1..56.

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

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.

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 *)

Crossrefs

Cf. A166243, A166244, A159843.

Sequence in context: A065104 A138645 A191060 * A250185 A063206 A063099

Adjacent sequences: A166243 A166244 A166245 * A166247 A166248 A166249

Keyword

nonn

Author

Max Alekseyev, Oct 10 2009

Status

approved