Let $p$ be a prime number $\geq 17$. This paper deals with the diophantine equation $a^3+b^3=c^p$. If we suppose that the Taniyama-Weil conjecture is true, we get a criterion that often allows one to prove that this equation has no nonzero integer solution with $a$, $b$ and $c$ coprime. In particular, we verify that this is the case if $p$ is < 10000.

Soit $p$ un nombre premier $\geq 17$. Cet article concerne l'équation diophantienne $a^3+b^3=c^p$. En supposant que la conjecture de Taniyama-Weil soit vraie, nous démontrons un critère permettant souvent de prouver que cette équation n'a pas de solution en nombres entiers non nuls $a$, $b$ et $c$ dans $\Z$ premiers entre eux. Nous vérifions en particulier que tel est bien le cas si $p$ est < 10000.

This paper develops an approach to the evaluation of Euler sums that involve harmonic numbers, either linearly or nonlinearly. We give explicit formulæ for several classes of Euler sums in terms of Riemann zeta values. The approach is based on simple contour integral representations and residue computations.

If $P(x_1$,\,\dots,\,$x_n)$ is a polynomial with integer coefficients, the Mahler measure $M(P)$ of $P$ is defined to be the geometric mean of $|P|$ over the $n$-torus $\T ^n$. For $n = 1$, $M(P)$ is an algebraic integer, but for $n$\raise.5pt\hbox{\footnotesize\mathversion{bold}${}>{}$}$1$, there is reason to believe that $M(P)$ is usually transcendental. For example, Smyth showed that $\log M(1+x+y)=L'(${\mathversion{normal}$\chi$}$_{-3}$,$\,{-}1)$, where {\mathversion{normal}$\chi$}$_{-3}$ is the odd Dirichlet character of conductor $3$. Here we will describe some examples for which it appears that $\log M(P(x$,$\,y)) = r@@L'(E$,$\,0)$, where $E$ is an elliptic curve and $r$ is a rational number, often either an integer or the reciprocal of an integer. Most of the formulas we discover have been verified numerically to high accuracy but not rigorously proved.