Journal of Integer Sequences, Vol. 13 (2010), Article 10.2.1

On the Multiplicative Order of an Modulo n


Jonathan Chappelon
Université Lille Nord de France
F-59000 Lille
France

Abstract:

Let n be a positive integer and αn be the arithmetic function which assigns the multiplicative order of an modulo n to every integer a coprime to n and vanishes elsewhere. Similarly, let βn assign the projective multiplicative order of an 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 n1 and n2 with the same square-free part, there exists a relationship between the functions αn1 and αn2 and between the functions βn1 and βn2. This allows us to reduce the determination of αn and βn 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/nZ 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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