This article is concerned with the large-prime variations of the multipolynomial quadratic sieve factorization method: the PMPQS (one large prime) and the PPMPQS (two). We present the results of many factorization runs with the PMPQS and PPMPQS on SGI workstations and on a Cray C90 vector computer. Experiments show that for our Cray C90 implementations PPMPQS beats PMPQS for numbers of more than 80 digits, and that this crossover point goes down with the amount of available central memory.

For PMPQS we give a formula to predict the total running time based on a short test run. The accuracy of the prediction is within 10\% of the actual running time. For PPMPQS we do not have such a formula. Yet in order to provide measurements to help determining a good choice of the parameters in PPMPQS, we factored many numbers. In addition we give an experimental prediction formula for PPMPQS suitable if one wishes to factor many large numbers of about the same size.

It is known by a result of Langlands and Weil that one can associate to each representation of the absolute Galois group of $\Q$ with odd determinant and octahedral type a newform of weight one. Using the work of Bayer and Frey, we provide a method for constructing such newforms. The calculation of the coefficients of their Fourier expansions at infinity can then be computed so as to provide tables. The case of forms of even level is studied in detail.

Un résultat de Langlands et Weil permet d'associer à toute représentation galoisienne de type octaédral et de déterminant impair du groupe de Galois absolu de $\Q$ une forme primitive de poids $1$. En nous appuyant sur les travaux de Bayer et Frey, nous proposons une méthode de construction de formes primitives basée sur ce résultat. Le calcul des coefficients de leur développement de Fourier à l'infini, que nous avons implémenté sur machine, permet la construction de tables. Le cas des formes de niveau pair est étudié avec précision.

We describe an elementary method for computing isolated values of $M(x)=\sum_{n \leq x} \mu(n)$, where $\mu$ is the Möbius function. The complexity of the algorithm is $O(x^{2/3}(\log \log x)^{1/3})$ time and $O(x^{1/3}(\log \log x)^{2/3})$ space. Certain values of $M(x)$ for $x$ up to $10^{16}$ are listed: for instance, $M(10^{16})=-3195437$.

In this paper we study growth functions of automatic and hyperbolic groups. In addition to standard growth functions, we also want to count the number of finite graphs isomorphic to a given finite graph in the ball of radius $n$ around the identity element in the Cayley graph. This topic was introduced to us by K. Saito [1991]. We report on fast methods to compute the growth function once we know the automatic structure. We prove that for a geodesic automatic structure, the growth function for any fixed finite connected graph is a rational function. For a word-hyperbolic group, we show that one can choose the denominator of the rational function independently of the finite graph.

A latin square is an $n\times n$ array of $n$ symbols in which each symbol appears exactly once in each row and column. Regarding each symbol as a variable and taking the determinant, we get a degree-$n$ polynomial in $n$ variables. Can two latin squares $L,M$ have the same determinant, up to a renaming of the variables, apart from the obvious cases when $L$ is obtained from $M$ by a sequence of row interchanges, column interchanges, renaming of variables, and transposition? The answer was known to be no if $n\le7$; we show that it is yes for $n=8$. The latin squares for which this situation occurs have interesting special characteristics.

Equivariant polynomial functions with the symmetries of the $n$-cube are completely determined in terms of permutations of exponents. Strategies for random searches of linear combinations of these functions are described and used to generate interesting examples of attractors. These attractors have symmetries that are an admissible subgroup of the symmetries of the square, cube and 4-cube. A central projection of the 4-cube with partial inversion is used for the illustrations of attractors in four dimensions.