The PARIS cluster coupled to the BaFPro electronic module: data analysis from the NRF experiment at the γ\gammaELBE facility

International audience; The first cluster of the constructed PARIS calorimeter was assembled and tested atthe ELBE facility at HZDR, Dresden, Germany. The experiment was aimed at the evaluationof the performance of each detector separately as well as the whole PARIS cluster with discrete$\gamma$-ray energies seen by the PARIS ranging up to 8.9 MeV. As the detectors use phoswichconfiguration, with 2'' x 2'' x 2'' LaBr3(Ce) crystal coupled to 2'' x 2'' x 6'' NaI(Tl) one, greatcare must be taken during the data analysis process to obtain the best possible values for energyresolution. Two algorithms for data transformation from matrices created with slow vs fastpulse shaping to energy spectra were tested from which one was chosen for further analysis. Analgorithm for adding back energies of $\gamma$-rays scattered inside the cluster was prepared, as well.Energy resolution for $\gamma$-rays in 2–8 MeV range was estimated and is presented in this paper

On a Few Diophantine Equations Related to Fermat’s Last Theorem

Abstract. We combine the deep methods of Frey, Ribet, Serre and Wiles with some results of Darmon, Merel and Poonen to solve certain explicit diophantine equations. In particular, we prove that the area of a primitive Pythagorean triangle is never a perfect power, and that each of the equations X4 −4Y 4 = Z p, X4 + 4Y p = Z2 has no non-trivial solution. Proofs are short and rest heavily on results whose proofs required Wiles ’ deep machinery.

On fixed divisors of the values of the minimal polynomials over Z of algebraic numbers

Let $K$ be a number field of degree $n$, $A$ be its ring of integers, and $A_n$ (resp. $K_n$) be the set of elements of $A$ ( resp. $K$) which are primitive over $\mathbb Q$. For any $\gamma \in {K_n}$, let $F_{\gamma} (x)$ be the unique irreducible polynomial in $\mathbb Z[x]$, such that its leading coefficient is positive and $F_{\gamma} ({\gamma}) = 0$. Let $i(\gamma)=\gcd_{x\in\mathbb Z}F_{\gamma}(x)$, i(K)=\lcm_{\theta\in{A_n}}i(\theta) and \hat{\imath}(K) = \lcm_{\gamma\in{K_n}}i(\gamma). For any $\gamma \in {K_n}$, there exists a unique pair $(\theta,d)$, where $\theta\in A_n$ and $d$ is a positive integer such that $\gamma=\theta/d$ and $\theta\not\equiv 0\pmod{p}$ for any prime divisor $p$ of $d$. In this paper, we study the possible values of $\nu_{p}(d)$ when $p | i(\gamma)$. We introduce and study a new invariant of $K$ defined using $\nu_{p}(d)$, when $\gamma$ describes $K_n$. In the last theorem of this paper, we establish a generalisation of a theorem of MacCluer

ON FIXED DIVISORS OF THE MINIMAL POLYNOMIALS OVER Z OF ALGEBRAIC NUMBERS

International audienceLet K be a number field of degree n, A be its ring of integers, and A n (resp. K n) be the set of elements of A (resp. K) which are primitive over Q. For any γ ∈ K n , let F γ (x) be the unique irreducible polynomial in Z[x], such that its leading coefficient is positive and F γ (γ) = 0. Let i(γ) = gcd x∈Z F γ (x), i(K) = lcm θ ∈A n i(θ) andî(K) = lcm γ∈K n i(γ). For any γ ∈ K n , there exists a unique pair (θ , d), where θ ∈ A n and d is a positive integer such that γ = θ /d and θ ≡ 0 (mod p) for any prime divisor p of d. In this paper, among other results, we show that i(K) andî(K) have the same prime factors and we study the possible values of ν p (d) when p|i(γ). We introduce and study a new invariant of K defined using ν p (d), when γ describes K n. In the last theorem of this paper, we establish a generalisation of a theorem of MacCluer. From Lemma 2.1 we see that any prime factor p of c(γ) divides d(γ). Summarizing the relations between c(γ) and d(γ), we have : Remark 2.1. Let K be a number field of degree n and γ ∈ K n , then d(γ) and c(γ) have the same prime factors and for any prime p, we have ν p (d(γ)) ≤ ν p (c(γ)) ≤ nν p (d(γ))

A new proof of the Carlitz-Lutz theorem

DOI: 10.1017/S000497271900072