Four primality testing algorithms

In this expository paper we describe four primality tests. The first test is very efficient, but is only capable of proving that a given number is either composite or 'very probably' prime. The second test is a deterministic polynomial time algorithm to prove that a given numer is either prime or composite. The third and fourth primality tests are at present most widely used in practice. Both tests are capable of proving that a given number is prime or composite, but neither algorithm is deterministic. The third algorithm exploits the arithmetic of cyclotomic fields. Its running time is almost, but not quite polynomial time. The fourth algorithm exploits elliptic curves. Its running time is difficult to estimate, but it behaves well in practice.Comment: 21 page

Adsorption of Oppositely Charged Polyelectrolytes onto a Charged Rod

The adsorption of highly charged flexible polycations and polyanions on a charged cylindrical substrate is investigated by means of Monte Carlo (MC) simulations. A detailed structural study, including monomer and fluid charge distributions, is provided. The influence of a short range attraction between the polycations and the negatively charged substrate is also considered. We demonstrate that the building up of multilayer structures is highly prohibited mainly due to the high entropy penalty stemming from the low dimensionality of the substrate at strong curvature.Comment: 18 pages - 7 (main) EPS figs - revtex4 - to appear in J. Chem. Phy