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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
Keywords:
Zsigmondy prime
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References:
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