2-Selmer groups and the Birch-Swinnerton-Dyer Conjecture for the congruent number curves
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Since the discovery of the utility of the numbers, the human being tried to differentiate them. We decide between them according to whether they are even or odd. Or, according to the fact that they are prime or composite. A natural number n >1 is called a ...
We describe how we reached a new factoring milestone by completing the first special number field sieve factorization of a number having more than 1024 bits, namely the Mersenne number 21039 -1. Although this factorization is orders of magnitude ...
The generation of prime numbers underlies the use of most public-key schemes, essentially as a major primitive needed for the creation of key pairs or as a computation stage appearing during various cryptographic setups. Surprisingly, despite decades of in ...
Let p be an odd prime and zetap be a primitive pth root of unity over smallBbbQ. The Galois group G of K:=smallBbbQ(zetap) over smallBbbQ is a cyclic group of order p−1. The integral group ring smallBbbZ[G] contains the Stickelberger idea ...
We consider the following integer feasibility problem: Given positive integer numbers a0, a1,&mellip;,an with gcd(a1,&mellip;,an) = 1 and a = (a1,&mellip;,an), does there exist a ...
Let d(n) denote Dirichlet's divisor function for positive integer numbers. This work is primarily concerned with the study of We are interested, in the error term where Ρ3 is a polynomial of degree 3 ; more precisely xΡ3(log x) is the residue of in s = 1. ...
Computations of irregular primes and associated cyclotomic invariants were extended to all primes up to twelve million using multisectioning/convolution methods and a novel approach which originated in the study of Stickelberger codes (Shokrollahi (1996)). ...
Factoring, finding a non-trivial factorization of a composite positive integer, is believed to be a hard problem. How hard we think it is, however, changes almost on a daily basis. Predicting how hard factoring will be in the future, an important issue for ...
The polynomial time algorithm of Lenstra, Lenstra, and Lovász [15] for factoring integer polynomials and variants thereof have been widely used to show that various computational problems in number theory have polynomial time solutions. Among them is the p ...
Let K/Fq be an elliptic function field. For every natural number n we determine the number of prime divisors of degree n of K/Fq which lie in a given divisor class of K ...
