Sylvester’s Problem and Mock Heegner Points
نویسندگان
چکیده
We prove that if p ≡ 4, 7 (mod 9) is prime and 3 is not a cube modulo p, then both of the equations x + y = p and x + y = p have a solution with x, y ∈ Q.
منابع مشابه
Heegner points and Sylvester’s conjecture
We consider the classical Diophantine problem of writing positive integers n as the sum of two rational cubes, i.e. n = x3 + y3 for x, y ∈ Q. A conjecture attributed to Sylvester asserts that a rational prime p > 3 can be so expressed if p ≡ 4, 7, 8 (mod 9). The theory of mock Heegner points gives a method for exhibiting such a pair (x, y) in certain cases. In this article, we give an expositor...
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Elliptic curves over Q are equipped with a systematic collection of Heegner points arising from the theory of complex multiplication and defined over abelian extensions of imaginary quadratic fields. These points are the key to the most decisive progress in the last decades on the Birch and Swinnerton-Dyer conjecture: an essentially complete proof for elliptic curves over Q of analytic rank ≤ 1...
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The famous Sylvester’s problem is: Given finitely many noncollinear points in the plane, do they always span a line that contains precisely two of the points? The answer is yes, as was first shown by Gallai in 1944. Since then, many other proofs and generalizations of the problem appeared. We present two new proofs of Gallai’s result, using the powerful method of allowable sequences.
متن کاملPreface Henri Darmon
Modular curves and their close relatives, Shimura curves attached to multiplicative subgroups of quaternion algebras, are equipped with a distinguished collection of points defined over class fields of imaginary quadratic fields and arising from the theory of complex multiplication: the so-called Heegner points. It is customary to use the same term to describe the images of degree zero divisors...
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تاریخ انتشار 2017