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Modular zero divisors of longest exponentiation cycle. (English) Zbl 1335.11004
Summary: We show that the sequence \(w^k \bmod n\), given that \(\gcd(w, n) > 1\), can reach a maximal cycle length of \(\varphi(n)\) if and only if \(n\) is twice an odd prime power, \(w\) is even, and \(w\) is a primitive root modulo \(n/2\).
MSC:
11A05 Multiplicative structure; Euclidean algorithm; greatest common divisors
11A07 Congruences; primitive roots; residue systems
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