Following a 70-year old suggestion of Paul Lévy, a condition is formulated for the regularity of growth of real functions. The condition, which is quite explicit, makes use of the iterates of the function and solutions of Abel's functional equation, and is well adapted to a computer testing.

Numerous computer experiments reveal interesting properties of the proposed regularity criterion.

We investigate the close relationship between minimal surfaces in Euclidean three-space and surfaces of constant mean curvature 1 in hyperbolic three-space. Just as in the case of minimal surfaces in Euclidean three-space, the only complete connected embedded surfaces of constant mean curvature 1 with two ends in hyperbolic space are well-understood surfaces of revolution: the catenoid cousins.

In contrast to this, we show that, unlike the case of minimal surfaces in Euclidean three-space, there do exist complete connected immersed surfaces of constant mean curvature 1 with two ends in hyperbolic space that are not surfaces of revolution: the genus-one catenoid cousins. These surfaces are of interest because they show that, although minimal surfaces in Euclidean three-space and surfaces of constant mean curvature 1 in hyperbolic three-space are intimately related, there are essential differences between these two sets of surfaces. The proof we give of existence of the genus-one catenoid cousins is a mathematically rigorous verification that the results of a computer experiment are sufficiently accurate to imply existence.

The algebraic number fields of degree 6 having Galois group $S_5$ and minimum discriminant are determined for signatures (0,\,3), (2,\,2) and (6,\,0). The fields $F_0$, $F_2$, $F_6$ are generated by roots of $f_0(t) = t^6 + 3 t^4 + 2 t^3 + 6 t^2 + 1$, $\,f_2(t) = t^6 - 2 t^4 + 12 t^3 - 16 t + 8$, and $f_6(t) = t^6 - 18 t^4 + 9 t^3 + 90 t^2 - 70 t - 69$ respectively. Each of these fields is unique up to isomorphism. This completes the enumeration of primitive sextic fields with minimum discriminant for all possible combinations of Galois group and signature.

Extended gcd calculation has a long history and plays an important role in computational number theory and linear algebra. Recent results have shown that finding optimal multipliers in extended gcd calculations is difficult. We present an algorithm which uses lattice basis reduction to produce small integer multipliers $x_1,\dots,x_m$ for the equation $s=\gcd{(s_1,\dots,s_m)}=x_1s_1+\cdots+x_ms_m$, where $s_1,\dots,s_m$ are given integers. The method generalises to produce small unimodular transformation matrices for computing the Hermite normal form of an integer matrix.

We say that an integer $n$ is abundant if the sum of the divisors of $n$ is at least $2n$. It has been known [wall71] that the set of abundant numbers has a natural density $A(2)$ and that $0.244 < A(2) < 0.291$. We give the sharper bounds $0.2474 < A(2) < 0.2480$.

Recall the well-known $3x+1$ conjecture: if $T(n)=(3n+1)$/$2$ for $n$ odd and $T(n)=n$/$2$ for $n$ even, repeated application of $T$ to any positive integer eventually leads to the cycle $$\{1\to2\to1\}\hbox{.}$$ We study a natural generalization of the function $T$, where instead of $3n+1$ one takes $3n+d$, for $d$ equal to -1 or to an odd positive integer not divisible by 3. With this generalization new cyclic phenomena appear, side by side with the general convergent dynamics typical of the $3x+1$ case. Nonetheless, experiments suggest the following conjecture: For any odd $d \ge -1$ not divisible by 3 there exists a finite set of positive integers such that iteration of the $3x+d$ function eventually lands in this set.

Along with a new boundedness result, we present here an improved formalism, more clear-cut and better suited for future experimental research.

We describe the theoretical background for a computer program that recognizes all closed orientable three-manifolds up to complexity 8. The program can treat also nonclosed three-manifolds and manifolds of greater complexity, but without necessarily succeeding in recognizing them.

The modular curve $\Xnd{11}$ (or $X_{\mathrm{nonsplit}}(11)$) classifies the elliptic curves $E$ such that Galois acts on the group of 11-torsion points of $E$ through the normaliser of a non-split Cartan subgroup. It is known that this curve has genus 1. We give here a parametrisation of this curve by a certain elliptic curve over $\Q$ with conductor 121. As an application, we give explicit examples of couples of elliptic curves over $\Q$ nonisogenous over $\Q$ but giving symplectically isomorphic modulo 11 Galois representations.

La courbe modulaire $\Xnd{11}$ classifie les courbes elliptiques $E$ telles que l'image de la représentation de Galois dans le groupe des points de 11-torsion de $E$ soit contenue dans le normalisateur d'un sous-groupe de Cartan non d'éployé. On sait que $\Xnd{11}$ est de genre 1. Nous donnons ici une paramétrisation de cette courbe par une certaine courbe elliptique sur $\Q$, de conducteur 121. Nous en déduisons des exemples explicites de couples de courbes elliptiques sur $\Q$ non isogènes sur $\Q$ mais donnant des représentations de Galois modulo 11 symplectiquement isomorphes.