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A166246 Primes representable as the sum of two rational cubes. 6

%I

%S 2,7,13,17,19,31,37,43,53,61,67,71,79,89,97,103,107,127,139,151,157,

%T 163,179,193,197,211,223,229,233,241,251,269,271,277,283,313,331,337,

%U 349,359,367,373,379,397,409,421,431,433,439,449,457,463,467,499,503,521

%N Primes representable as the sum of two rational cubes.

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

%C Also, the prime elements of A020898.

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

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

%F (i) p = 2;

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

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

%t (* 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 *)

%Y Cf. A166243, A166244, A159843.

%K nonn

%O 1,1

%A _Max Alekseyev_, Oct 10 2009

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Last modified June 20 10:03 EDT 2021. Contains 345162 sequences. (Running on oeis4.)