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Which primes are sums of two cubes?


Zagier,  Don
Max Planck Institute for Mathematics, Max Planck Society;

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Rodriguez Villegas, F., & Zagier, D. (1995). Which primes are sums of two cubes? In Number theory. Fourth conference of the Canadian Number Theory Association, July 2-8, 1994, Dalhousie University, Halifax, Nova Scotia, Canada (pp. 295-306). Providence, RI: American Mathematical Society.

Cite as: http://hdl.handle.net/21.11116/0000-0004-38FD-1
Let L(Ep, s) be the L-series of the elliptic curve Ep: x^3+ y^3= pz^3, p being a rational prime p\equiv 1\pmod 9; let cp= \sqrt 3 Γ (1/3)^3 (2π)^-1 p^-1/3. Then L(Ep, 1)= cp Sp with Sp\in \bbfZ. The authors give several formulae for Sp. They prove, in particular, that Sp= \text tr αp, and Sp= (\text tr βp)^2 for some algebraic integers αp, βp; thus Sp is always a square, as expected. One of their formulae implies that Sp= 0\iff p\mid fn(p) (0), where fn (t) is a certain sequence of polynomials defined by a simple recursion relation, n(p):= 2(p- 1)/3. According to the Birch and Swinnerton-Dyer conjecture, Sp =0 if and only if Ep (\bbfQ)\ne 0; therefore the authors' formulae give a (conjectural) answer to the question posed in the title of their paper.