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A166246 Primes representable as the sum of two rational cubes. 6
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 (list; graph; refs; listen; history; text; internal format)
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.

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-Fran├žois 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

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Last modified October 23 07:11 EDT 2019. Contains 328336 sequences. (Running on oeis4.)