Cusanus Institut

A prime number p is called elite if only finitely many Fermat numbers 2 2 n + 1 are quadratic residues modulo p. So far, all 21 elite primes less than 250 billion were known, together with 24 larger items. We completed the systematic search up to the range of 2.5 · 10 12 finding six more such numbers. Moreover, 42 new elites larger than this bound were found, among which the largest has 374 596 decimal digits. A survey on the knowledge about elite primes together with some open problems and conjectures are presented.

Le sujet de l’article est la situation actuelle de la recherche internationale sur le Cusain et ses perspectives pour l’avenir. Au cours du xxe siècle, la recherche internationale sur Nicolas de Cues a connu une expansion énorme. L’édition historique-critique de son œuvre est achevée. Aujourd’hui, il existe des centres sur le Cusain pas seulement en Allemagne, au Japon et aux États-Unis, mais aussi aux Pays-Bas, en Italie, en Argentine, en Russie et en France. à l’échelle mondiale, beaucoup de thèses de doctorat sont écrites sur Nicolas de Cues, sans parler des nombreux articles le concernant. à l’occasion du 25e anniversaire de l’American Cusanus Society en mai 2008 à Kalamazoo, le philosophe américain Clyde Lee Miller a étudié la recherche sur Nicolas de Cues sous quatre angles différents que l’article reprend pour esquisser les futures perspectives de la recherche sur le Cusain.

Denote by ρ(k) the smallest prime number with digital sum k (not a multiple of 3). Richard K. Guy asked whether the congruences ρ(k) ≡ 99 (mod 100) and ρ(k) ≡ 999 (mod 1000) hold for all k > 38, respectively k > 59. Counterexamples to this are given, inter alia, for k = 86 and k = 104. Moreover, several open problems and conjectures about ρ(k) are discussed.

Nicholas of Cusa authored three writings dealing mainly with Islam: ‘De pace fidei’ dating from September 1453‚ the letter addressed to John of Segovia from December 29, 1454 and ‘Cribratio Alkorani’‚ 1460/61. In these three writings an interesting development can be seen. Cusa ignores the anti-Christian side of Islam in ‘De pace fidei’‚ presenting it as a mere misunderstanding. In his letter to John of Segovia he already implies that he is aware of the ambivalence of Islam (visible in his remark about the hermeneutic of the Koran), and he goes on to discuss this extensively in ‘Cribratio Alkorani’. The uniformity of the interpretation of Islam and the Koran suffers accordingly.