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14 Mars Jour transcendant

Aujourd’hui c’est le PI Day . Le jour anniversaire du nombre PI soit 3,14159 …
En effet le 14 Mars (3 14) à 1H 59 minutes ce sera l’anniversaire de ce nombre transcendant.

Le nombre Pi a bien sûr son site : ici

En mathématiques, un nombre transcendant est un nombre réel ou complexe qui n’est racine d’aucune équation polynomiale :

a_n~x^n + a_{n-1}~x^{n-1} + \cdots + a_1~x^1 + a_0 = 0\,

n \ge 1\, et les coefficients a_i\, sont des nombres entiers (donc des rationnels), dont au moins l’un an est non nul. Un nombre réel ou complexe est donc transcendant si et seulement s’il n’est pas algébrique.

Si on sait démontrer que des  nombres sont transcendants comme e et Pi, certains nombres résistent comme la constante de Catalan, encore un Catalan me direz vous.

En mathématiques, la constante de Catalan, nommée d’après le mathématicien Eugène Charles Catalan, est le nombre défini par :

K = \beta(2)=\sum_{n=0}^\infty \frac{(-1)^n}{(2n+1)^2}\simeq 0,91596559417721901505…,

? est la fonction beta de Dirichlet.

Ce n’est qu’en 2002 que Preda V. Mih?ilescu, un mathématicien roumain démontra  la conjecture de Catalan.

Deux nombres entiers consécutifs, autres que 8 et 9 ne peuvent être des puissances exactes; autrement dit: l’équation

x^p - y^q = 1 \,

dans laquelle les inconnues sont entières et positives, n’admet qu’une seule solution

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