On the ratio of the sum of divisors and Euler’s totient function II

We find the form of all solutions to ø(n) | σ(n) with three or fewer prime factors, except when the quotient is 4 and n is even

Computing the endomorphism ring of an ordinary elliptic curve over a finite field

We present two algorithms to compute the endomorphism ring of an ordinary elliptic curve E defined over a finite field F_q. Under suitable heuristic assumptions, both have subexponential complexity. We bound the complexity of the first algorithm in terms of log q, while our bound for the second algorithm depends primarily on log |D_E|, where D_E is the discriminant of the order isomorphic to End(E). As a byproduct, our method yields a short certificate that may be used to verify that the endomorphism ring is as claimed.Comment: 16 pages (minor edits

Generating elements of orders dividing p6 ± p5 + p4 ± p3 + p2 + p ± 1

In this paper we propose an algorithm for computing large primes p and q such that q divides p6 + p5 + p4 + p3 + p2 + p + 1 or p6 - p5 + p4 - p3 + p2 - p + 1. Such primes are the key parameters for the cryptosystem based on the 7th order characteristic sequences

The Brauer-Manin Obstruction and Sha[2].

We discuss the Brauer-Manin obstruction on del Pezzo surfaces of degree 4. We outline a detailed algorithm for computing the obstruction and provide associated programs in magma. This is illustrated with the computation of an example with an irreducible cubic factor in the singular locus of the defining pencil of quadrics (in contrast to previous examples, which had at worst quadratic irreducible factors). We exploit the relationship with the Tate-Shafarevich group to give new types of examples of Sha[2], for families of curves of genus 2 of the form y^2 = f(x), where f(x) is a quintic containing an irreducible cubic factor