Primality of closed path polyominoes

Distributed Primality Proving and the Primality of (23539+1)/3

We explain how the Elliptic Curve Primality Proving algorithm can be implemented in a distributed way. Applications are given to the certiication of large primes (more than 500 digits). As a result, we describe the successful attempt at proving the primality of the 1065-digit (2 3539 +1)=3, the rst ordinary Titanic prime.

Enumeration of generalized polyominoes

As a generalization of polyominoes we consider edge-to-edge connected nonoverlapping unions of regular k-gons. For n ≤ 4 we determine formulas for the number ak(n) of generalized polyominoes consisting of n regular k-gons. Additionally we give a table of the numbers ak(n) for small k and n obtained by computer enumeration. We finish with some open problems for k-polyominoes.

Primality testing

Helly Numbers of Polyominoes

We define the Helly number of a polyomino P as the smallest number h such that the h-Helly property holds for the family of symmetric and translated copies of P on the integer grid. We prove the following: (i) the only polyominoes with Helly number 2 are the rectangles, (ii) there does not exist any polyomino with Helly number 3, (iii) there exist polyminoes of Helly number k for any k 6= 1, 3.

Factoring & Primality

In this lecture we will discuss the problem of integer factorization and primality testing, two problems that have been the focus of a great amount of research over the years. These problems started receiving attention in the mathematics community far before the appearance of computer science, however the emergence of the latter has given them additional importance. Gauss himself wrote in 1801 ...