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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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