Chaumont, Alain; Leicht, Johannes; Müller, Tom; Reinhart, Andreas The continuing search for large elite primes. (English) Zbl 1178.11001 Int. J. Number Theory 5, No. 2, 209-218 (2009). A prime number \(p\) is called elite provided that only finitely many Fermat numbers \(F_n=2^{2^n}+1\) are quadratic residues modulo \(p\). The authors made a systematic search for elite primes smaller than \(2.5\cdot 10^{12};\) their search yielded 27 elite primes. This adds to a previous search which showed that there are 21 elite primes less than \(2.5\cdot 10^{11}\); [see A. Chaumont and T. Müller, J. Integer Seq. 9, No. 3, Article 06.3.8, 5 p., electronic only (2006; Zbl 1178.11002)]. In addition to this search, they also found 42 new elite Proth primes. A Proth prime is a prime of the form \(2^r\cdot h+1\) where \(r\geq 2\) and \(h\) is an odd number satisfying \(h<2^r.\) The largest of these 42 elite Proth primes has \(374596\) decimal digits. Reviewer: Michael Coons (Waterloo) Cited in 1 Document MSC: 11-04 Software, source code, etc. for problems pertaining to number theory 11A15 Power residues, reciprocity 11A41 Primes 11Y11 Primality Keywords:elite primes; Fermat numbers; elite Proth primes Citations:Zbl 1178.11002 PDFBibTeX XMLCite \textit{A. Chaumont} et al., Int. J. Number Theory 5, No. 2, 209--218 (2009; Zbl 1178.11001) Full Text: DOI References: [1] DOI: 10.1007/BF01298923 · Zbl 0584.10003 · doi:10.1007/BF01298923 [2] Chaumont A., J. Integer Seq. 9 [3] Dickson L. E., History of the Theory of Numbers 1 (2005) · Zbl 1214.11002 [4] L. Euler, Commentarii Academiae Scientiarum Imperialis Petropolitanae 6 (Imperial Academy of Sciences, St. Petersburg, 1739) pp. 103–107. [5] P. S. Fermat, Oeuvres 2 (Gauthier-Villars, Paris, 1894) p. 309. [6] DOI: 10.1007/978-0-387-26677-0 · doi:10.1007/978-0-387-26677-0 [7] Hardy G. H., An Introduction to the Theory of Numbers (1979) · Zbl 0423.10001 [8] DOI: 10.1007/978-0-387-21850-2 · doi:10.1007/978-0-387-21850-2 [9] DOI: 10.1006/jnth.2002.2782 · Zbl 1026.11011 · doi:10.1006/jnth.2002.2782 [10] DOI: 10.1080/10586458.2006.10128955 · Zbl 1132.11005 · doi:10.1080/10586458.2006.10128955 [11] Proth E., Comptes Rendus Acad. Sci. Paris 87 pp 926– [12] DOI: 10.1007/978-1-4612-0251-6 · doi:10.1007/978-1-4612-0251-6 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.