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The continuing search for large elite primes. (English) Zbl 1178.11001

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.

MSC:

11-04 Software, source code, etc. for problems pertaining to number theory
11A15 Power residues, reciprocity
11A41 Primes
11Y11 Primality

Citations:

Zbl 1178.11002
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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
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