Let Fq be a finite field with q elements, ψ a non-zero element of Fq, and n an integer ≥ 3 prime to q. The aim of this article is to show that the zeta function of the projective variety over Fq defined by Xψ : x n 1 + · · · + x n n − nψx1 . . . xn = 0 has, when n is prime and Xψ is non singular (i.e. when ψ 6= 1), an explicit decomposition in factors coming from affine varieties of odd dimensi...

Let \( \{F_n\}_{n\geq 0} \) be the sequence of Fibonacci numbers and let \(p\) a prime. For an integer \(c\) we write \(m_{F,p}(c)\) for number distinct representations as \(F_k-p^\ell\) with \(k\ge 2\) \(\ell\ge 0\). We prove that \(m_{F,p}(c)\le 4\).

contributions to Catalan's classical problem to show that 8,9 is the only pair of consecutive positive integers which are both perfect powers. The work of Professor Ko and the related work by others show that there is just one mathematical world and that there are no borders in this world. For any positive integer n let P (n); !(n); (n) denote the greatest prime factor of n, the number of disti...

The Atacama Large Millimeter/submillimeter Array (ALMA) serendipitously detected H$_2$O $J_{Ka, Kc} = 10_{2,9} - 9_{3,6}$ emission at 321 GHz in NGC 1052. This is the first submillimeter maser detection a radio galaxy and most luminous 321-GHz known to date with isotropic luminosity of 1090 $L_{\odot}$. line profile consists broad velocity component FWHM $= 208 \pm 12$ km s$^{-1}$ straddling sy...

We study the zeros (mod p) of the polynomial ßp(X) = ¿Zk(Bk/k)(Xp~l~k 1) for p an odd prime, where Bk denotes the k\\\ Bernoulli number and the summation extends over 1 < k < p — 2. We establish a reciprocity law which relates the congruence ßp(r) = 0 (mod p) to a congruence f (n) = 0 (mod r) for r a prime less than/7 and n e Z. The polynomial/ (x) is the irreducible polynomial over Q of the nu...

We continue our study on the elliptic curve discrete logarithm problem over finite extension fields. We show, among other results, the following two results: For sequences of prime powers (qi)i∈N and natural numbers (ni)i∈N with ni −→ ∞ and ni log(qi) −→ 0 for i −→ ∞, the discrete logarithm problem in the groups of rational points of elliptic curves over the fields Fqi i can be solved in subexp...

"This work studies the initial boundary value problem for Petrovsky equation with nonlinear damping \begin{equation*} \frac{\partial ^{2}u}{\partial t^{2}}+\Delta ^{2}u-\Delta u^{\prime} +\left\vert u\right\vert ^{p-2}u+\alpha g\left( u^{\prime }\right) =\beta f\left( u\right) \text{ in }\Omega \times \left[ 0,+\infty \right[, \end{equation*} where $\Omega $ is open and bounded domain $\mathbb{...

— Let F be a number field, abelian over Q, and fix a prime p 6= 2. Consider the cyclotomic Zp-extension F∞/F and denote Fn the n th finite subfield and Cn its group of circular units. Then the Galois groups Gm,n = Gal(Fm/Fn) act naturally on the Cm’s (for any m ≥ n >> 0). We compute the Tate cohomology groups Ĥ(Gm,n, Cm) for i = −1, 0 without assuming anything else neither on F nor on p.

The energy minima of systems made of an alanine molecule, a lithium cation and a various number of watermolecules have been determined with the help of quantum mechanical computations at the B3LYP DensityFunctional Theory level of computation and the standard 6-311++G (d,p) basis set. Several structures, close inenergy are found and the presence of one or two water molecules around the cation m...