Landau’s problems on primes

Pintz, János

Journal de théorie des nombres de Bordeaux, Tome 21 (2009) no. 2, pp. 357-404.

Au congrès international de Cambridge en 1912, Laudau dressa la liste de quatre problèmes de base sur les nombres premiers. Ces problèmes furent caractérisés dans son discours comme “inaccessibles en l’état actuel de la science”. Ces problèmes sont les suivants :

• (1) Existe-t-il une infinité de nombres premiers de la forme $n^2+1$ ?
• (2) La conjecture (binaire) de Goldbach, que chaque nombre pair supérieur à 2 est somme de deux nombres premiers.
• (3) La conjecture des nombres premiers jumeaux.
• (4) Existe-t-il toujours un nombre premier entre deux carrés consécutifs ?

Tous ces problèmes sont encore ouverts. Le travail présenté ici est un exposé des résultats partiels aux problèmes (2)–(4), avec une attention particuliere concernant les résultats récents de D. Goldston, C. Yıldırım et de l’auteur sur les petits écarts entre nombres premiers.

At the 1912 Cambridge International Congress Landau listed four basic problems about primes. These problems were characterised in his speech as “unattackable at the present state of science”. The problems were the following :

• (1) Are there infinitely many primes of the form $n^2+1$?
• (2) The (Binary) Goldbach Conjecture, that every even number exceeding 2 can be written as the sum of two primes.
• (3) The Twin Prime Conjecture.
• (4) Does there exist always at least one prime between neighbouring squares?

All these problems are still open. In the present work a survey will be given about partial results in Problems (2)–(4), with special emphasis on the recent results of D. Goldston, C. Yıldırım and the author on small gaps between primes.

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