The asymptotic frequency with which pairs of primes below x differ by some fixed integer is understood heuristically, although not rigorously, through the Hardy-Littlewood k-tuple conjecture. Less is known about the differences of consecutive primes. For all x between 1000 and $10^{12}$, the most common difference between consecutive primes is 6. We present heuristic and empirical evidence that 6 continues as the most common difference (jumping champion) up to about $x=1$.$7427\cdot10^{35}$, where it is replaced by 30. In turn, 30 is eventually displaced by 210, which is then displaced by 2310, and so on. Our heuristic arguments are based on a quantitative form of the Hardy-Littlewood conjecture. The technical difficulties in dealing with consecutive primes are formidable enough that even that strong conjecture does not suffice to produce a rigorous proof about the behavior of jumping champions.

We develop methods for the rapid computation of the regulator of a real quadratic congruence function field $K=k(x)({\sqrt{D}})$. By extending Shanks' infrastructure ideas in real quadratic number fields to real quadratic congruence function fields we obtain a baby step-giant step method for evaluating the regulator of K in $O( |D|\supfrac 14 )$ polynomial operations. We also show the existence of an effective algorithm which computes the regulator unconditionally in $O( |D|\supfrac 15 )$ polynomial operations. By implementing both methods on a computer, we found that the $O( |D|\supfrac 15 )$-algorithm tends to be far better than the baby step-giant step algorithm in those cases where the regulator exceeds $10^8$.

The elliptic logarithm method for the determination of all integral solutions of a given elliptic equation is discussed for equations with associated elliptic curve of moderately large rank. Major attention is given to the question of optimizing the choice of Mordell-Weil basis for the curves in question. A speculative argument suggests that for any curve of rank larger then 8 the calculations involved are unlikely to be feasible. The arguments are illustrated by examples of curves of rank 5, 6, 7, and 8, taken from the literature.

We have identified explicitly the dual of the hypersurface I5 defined as the unique quintic hypersurface in $P^5\funnyC$ invariant under the natural action of the Weyl group $W(\Ee)$ of the lattice $\Ee$.

This paper deals with the effective computation of normal forms, centre manifolds and first integrals in Hamiltonian mechanics. These calculations are very useful since they allow us, among other things, to give explicit estimates on the diffusion time and to compute invariant tori. The approach presented here is based on the algebraic manipulation of formal series with numerical coefficients for them. This, together with a very efficient software implementation, allows big savings in memory and execution time in comparison with the use of commercial algebraic manipulators. The algorithms are discussed together with their C/C++ implementations, and they are applied to some concrete examples from celestial mechanics.

We prove a formula for $\zeta(4n+3)$ discovered by Borwein and Bradley (Experimental Mathematics 6:3 (1997), 181-194).