For \(E\) an elliptic curve over a number field \(K\), we give a lower bound, conditional on the "parity conjecture." for the number of quadratic twists of \(E\) whose Mordell-Weil rank is at least two. The main tool is a sieve-theoretic estimate of the number of square-free values of a homogeneous
Modifications to the Number Field Sieve
β Scribed by Don Coppersmith
- Publisher
- Springer
- Year
- 1993
- Tongue
- English
- Weight
- 469 KB
- Volume
- 6
- Category
- Article
- ISSN
- 0933-2790
No coin nor oath required. For personal study only.
β¦ Synopsis
I-LLMP'I and Buhler et al. I'BLP], is a new routine for factoring integers. We present here a modification of that sieve. We use the fact that certain smoothness computations can be reused, and thereby reduce the asymptotic running time of the Number Field Sieve. We also give a way to precompute tables which will be useful for factoring any integers in a large range.
π SIMILAR VOLUMES
Ever since Viggo Brun's pioneering work, number theorists have developed increasingly sophisticated refinements of the sieve of Eratosthenes to attack problems such as the twin prime conjecture and Goldbach's conjecture. Ever since Gian-Carlo Rota's pioneering work, combinatorialists have found more