Landau’s problems on primes

Au congres international de Cambridge en 1912, Laudau dressa la liste de quatre problemes de base sur les nombres premiers. Ces problemes furent caracterises dans son discours comme "inaccessibles en l'etat actuel de la science". Ces problemes sont les suivants: (1) Existe-t-il une infinite de nombres premiers de la forme n 2 + 1 ? (2) La conjecture (binaire) de Goldbach, que chaque nombre pair superieur a 2 est somme de deux nombres premiers. (3) La conjecture des nombres premiers jumeaux. (4) Existe-t-il toujours un nombre premier entre deux carres consecutifs ? Tous ces problemes sont encore ouverts. Le travail presente ici est un expose des resultats partiels aux problemes (2)-(4), avec une attention particuliere concernant les resultats recents de D. Goldston, C. Yildirim et de l'auteur sur les petits ecarts entre nombres premiers.

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