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On the Multiplicative Order of ***a*^{n} Modulo *n*

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Jonathan Chappelon

Université Lille Nord de France

F-59000 Lille

France

**Abstract:**

Let *n* be a positive integer and α_{n} be
the arithmetic function which assigns the multiplicative order of
*a*^{n} modulo *n* to every integer *a*
coprime to *n* and vanishes elsewhere.
Similarly, let β_{n} assign
the projective multiplicative order of
*a*^{n} modulo *n* to every
integer *a* coprime to *n* and vanishes elsewhere. In this paper, we
present a study of these two arithmetic functions. In particular, we
prove that for positive integers *n*_{1} and
*n*_{2} with the same
square-free part, there exists a relationship between the functions
α_{n1} and
α_{n2}
and between the functions
β_{n1} and
β_{n2}.
This allows us to reduce the
determination of
α_{n}
and
β_{n}
to the case where *n* is
square-free. These arithmetic functions recently appeared in the
context of an old problem of Molluzzo, and more precisely in the study
of which arithmetic progressions yield a balanced Steinhaus triangle in
Z/*n*Z for *n* odd.

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Received October 16 2009;
revised version received January 20 2010.
Published in *Journal of Integer Sequences*, January 27 2010.

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