On the Multiplicative Order of an Modulo n
Université Lille Nord de France
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
and between the functions
This allows us to reduce the
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.
Full version: pdf,
Received October 16 2009;
revised version received January 20 2010.
Published in Journal of Integer Sequences, January 27 2010.
Journal of Integer Sequences home page