On établit la majoration |M(x)| 0.570591 \sqrt{ x }$, valable pour x appartenant à l'intervalle $\bigl[33 , 10^{12}\bigr]$, et on décrit les algorithmes qui ont permis de l'obtenir. On étudie également un algorithme de calcul de valeurs isolées de M(x), dont on montre que la complexité est en $O(x^{3/4} \log^{1/2}x)$ pour le temps de calcul, et en $O(x^{1/2})$ pour l'encombrement mémoire.

We establish the upper bound |M(x)| 0.570591 \sqrt{ x }$, valid for $x$ in the interval $\bigl[33 , 10^{12}\bigr]$, and we describe the algorithms used to obtain it. We also study an algorithm for the computation of isolated values of M(x), proving that its time and space complexities are $O(x^{3/4} \log^{1/2}x)$ and $O(x^{1/2})$, respectively.

On établit la majoration "intermédiare" $\mx\le \frac {1}{2360}x$, valable pour x appartenant à l'intervalle $\bigl[10^{12},\,2.2959\, 10^{16}\bigr]$, et d'autres majorations analogues. On établit ensuite la majoration asymptotique $\mx\le \frac {1}{2360}x$, valable pour tout $x\ge 617\, 973 $.

We establish the "intermediary" bound $\mx\le \frac {1}{2360}x$, valid for x in the interval $\bigl[10^{12},\,2.2959\, 10^{16}\bigr]$, and other similar upper bounds. Lastly, we establish the asymptotic upper bound $\mx\le \frac {1}{2360}x$, valid for every $x\ge 617\, 973 $.

After a brief review of partial results regarding CaseI 1 of Fermat's Last Theorem, we discuss the relationship between the number of points on Fermat's curve modulo a prime and the resultant $R_n$ of the polynomials $X^n - 1$ and $(-1-X)^n - 1$, called Wendt's determinant. The investigation of a conjecture about essential prime factors of $R_n$ (Conjecture 1.3) leads to a proof that Case 1 of Fermat's Last Theorem holds for any prime exponent $p>{}$2 such that $np+{}$1 is prime for some integer $n\le{}$500 not divisible by 3.

EDITOR'S NOTE: In addition to providing insight into Wendt's determinant, an object of interest in its own right, this paper belongs to a continuing line of investigations that may prove fruitful in spite of the recent announcement by Wiles of his proof of Fermat's Last Theorem. It is not unreasonable to hope for a more elementary proof than Wiles'.

We study the density of the set of real quadratic fields for which the norm of the fundamental unit equals $-$1 inside the set of real quadratic fields containing elements of norm $-$1. A conjectural density is derived from a single heuristic assumption, and experimental data supporting this assumption are given. We finally discuss how close one can get to proving such conjectural densities.

Nous construisons 53 corps quadratiques imaginaires ayant un 11-rang égal í 3, et sept corps quadratiques réels de 11-rang égal à 2. Ce sont, à notre connaissance, les premiers exemples de tels corps.

We construct 53 imaginary quadratic fields of 11-rank equal to 3, and seven quadratic fields of 11-rank equal to 2. These appear to be the first examples of such fields.