We study the rational points on $X_0^+(p) = X_0(p) / W_p$. It is known that there are rational points corresponding to cusps and elliptic curves with complex multiplication (CM). We use computational methods to exhibit exceptional rational points on $X_0^+(p)$ for p = 73, 103, 137, 191 and 311. We also provide the j-invariants of the corresponding non-CM quadratic $\funnyQ$-curves.

On décrit, essentiellement pour les polynômes du second degré, deux modèles heuristiques probabilistes donnant le nombre k de valeurs premières prises par ces polynômes sur un intervalle de n valeurs consécutives de la variable. Le premier modèle est consacré au cas k=n et fait apparaître un "mur de Schinzel", d'ordre de grandeur $n^n$, en-dessous duquel l'évènement "n valeurs premières pour n valeurs consécutives de la variable" est statistiquement exceptionnel, et au-dessus duquel il est statistiquement fréquent s'il n'y a pas d'obstruction arithmétique. Le deuxième modèle est consacré au cas k<n et est fond sur une loi binomiale avec avec une probabilité qui dérive de la constante de Hardy-Littlewood. Pour ce modèlle comme pour le précédent, les expérimentations numériques présentent une concordance satisfaisante. Ce travail est complété par une table des meilleurs polynômes trouvés jusqu'à n=40338. Enfin, on donne des résultats numériques pour les polynômes de degré 3, 4 et 5.

We describe two heuristic probabilistic models that give the number $k$ of prime values of quadratic polynomials on an interval of $n$ consecutive values of the variable. The first model is devoted to the case $k=n$ and reveals a "Schinzel barrier", of size $n^n$: below this barrier, the event "n prime values for n consecutive values of the variable" is statistically exceptional, and above, it is statistically frequent provided that there is no arithmetic obstruction. The second model is devoted to the case k<n and is based on a binomial law with a probability deriving from the Hardy-Littlewood constant. For both models, numerical experiments show a satisfactory similarity. This work is illustrated with a list of the best polynomials found for $n\leq40338$. We also give some numerical results for polynomials of degree 3, 4 and 5.

We present the first implementation of sieving techniques in the context of function fields. More precisely, we compute in class groups of quadratic congruence function fields by combining the algorithm of Hafner and McCurley with sieving ideas known from factoring. We apply our methods to the computation of generators and relations of the Jacobian variety of hyperelliptic curves over finite fields.

The algorithms introduced here were implemented in C++ with the help of LEDA and LiDIA. We provide examples of running times and comparisons with earlier algorithms.

Cet article décrit un algorithme nouveau et efficace pour calculer des polynômes caractéristiques d'endomorphismes dans un algèbre quotient en utilisant les modules de Cauchy. Cet algorithme est ensuite utilisé pour calculer des résolvantes et des multi-résolvantes absolues, outils de base de la théorie des corps et donc aussi de la théorie de Galois constructive.

We give a new and efficient algorithm to compute some characteristic polynomials using Cauchy modules. This algorithm is used for the computation of absolute resolvents and multi-resolvents which are essential tools in constructive Galois theory.

From a modern theta-function identity of G. E. Andrews we derive new representations for the celebrated Madelung constant and various of its analytic relatives. The method leads to connections with the modern theory of multiple zeta sums, generates an apparently entire "$\eta$ series" representation, and, for the Madelung constant in particular, yields a finite-integral representation. These analyses suggest variants of the Andrews identity, leading in turn to number-theoretical results concerning sums of three squares.

The Blasius function, denoted by f, is the solution to a simple nonlinear boundary layer problem, a third order ordinary differential equation on $x \in [0, \infty]$. In this work, we calculate several numerical constants, such as the second derivative of f at the origin and the two parameters of the linear asymptotic approximation to f, to at least eleven digits. Although the Blasius function is unbounded, we nevertheless derive an expansion in rational Chebyshev functions $\TL_{j}$ which converges exponentially fast with the truncation, and tabulate enough coefficients to compute f and its derivatives to about nine decimal places for all positive real x. The power series of f has a finite radius of convergence, ut the Euler-accelerated expansion is apparently convergent for all real x. We show that the singularities, which are first order poles to lowest order, have an infinite series of cosine-of-a-logarithm corrections. Lastly, we chart the behavior of f in the complex plane and conjecture that all singularities lie within three narrow sectors.

Using Gröbner basis algorithms in MAGMA we find necessary and sufficient conditions for a polynomial of degree 6 over any field to be the tensor product of two polynomials, one of degree 3 and one of degree 2.

We characterize all cyclotomic polynomials of even degree with coefficients restricted to the set $\{+1,-1\}$. In this context a cyclotomic polynomial is any monic polynomial with integer coefficients and all roots of modulus 1. \textit{Inter alia} we characterize all cyclotomic polynomials with odd coefficients.

The characterization is as follows. A polynomial $P(x)$ with coefficients $\pm1$ of even degree $N{-}1$ is cyclotomic if and only if

$$% P(x)=\pm\funnyPhi_{p_1}(\pm x)@\funnyPhi_{p_2}(\pm x^{p_1})\cdots \funnyPhi_{p_r}(\pm x^{p_1p_2\cdots p_{r-1}}) $$%,

where $N=p_1p_2\cdots p_r$ and the $p_i$ are primes, not necessarily distinct, and where $\funnyPhi_{p}(x):= (x^p-1)/(x-1)$ is the $p$-th cyclotomic polynomial.

We conjecture that this characterization also holds for polynomials of odd degree with $\pm 1$ coefficients. This conjecture is based on substantial computation plus a number of special cases.

Central to this paper is a careful analysis of the effect of Graeffe's root squaring algorithm on cyclotomic polynomials.

We study arithmetic progressions in the x-coordinates of rational points on elliptic curves. An infinite family of elliptic curves is found, each containing an arithmetic progression of length 8.