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On large Zsigmondy primes. (English) Zbl 0639.10007
Let \(a>1\), \(m>1\) be integers. A Zsigmondy prime for (a,m) is a prime p such that m is the smallest \(i\geq 1\) for which p divides \(a^ i-1\). Such primes exist, except in certain explicitly known cases [cf. K. Zsigmondy, Zur Theorie der Potenzreste, Monatsh. Math. Phys. 3, 265-284 (1892)]. A Zsigmondy prime p satisfies \(p\equiv 1 (mod m)\), so that \(p\geq m+1\). Such a prime p is called large if either \(p>m+1\) or \(p^ 2\) divides \(a^ m-1.\) In this paper all pairs (a,m) are listed for which no large Zsigmondy prime exists.
Reviewer: R.J.Stroeker

11A15 Power residues, reciprocity
11A07 Congruences; primitive roots; residue systems
Zsigmondy prime
Full Text: DOI
[1] Emil Artin, The orders of the linear groups, Comm. Pure Appl. Math. 8 (1955), 355 – 365. · Zbl 0065.01204
[2] Geo. D. Birkhoff and H. S. Vandiver, On the integral divisors of \?\(^{n}\)-\?\(^{n}\), Ann. of Math. (2) 5 (1904), no. 4, 173 – 180. · JFM 35.0205.01
[3] W. Feit, Extensions of cuspidal characters of \( {\text{G}}{{\text{L}}_m}\left( q \right)\) (to appear).
[4] W. Feit and G. Seitz (to appear).
[5] K. Zsigmondy, Zur Theorie der Potenzreste, Monatsh. Math. Phys. 3 (1892), no. 1, 265 – 284 (German). · JFM 24.0176.02
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