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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

##### MSC:
 11A15 Power residues, reciprocity 11A07 Congruences; primitive roots; residue systems
Zsigmondy prime
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##### References:
 [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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