Can the Arithmetic Derivative be Defined on a Non-Unique Factorization Domain?
Pentti Haukkanen, Mika Mattila, and Jorma K. Merikoski
School of Information Sciences
FI-33014 University of Tampere
School of Applied Educational Science and Teacher Education
University of Eastern Finland
P.O.Box 86, FI-57101 Savonlinna
its arithmetic derivative n
' is defined as follows: (i)
0'=1'=(-1)'=0. (ii) If
are primes (some of them possibly equal), then
An analogous definition can be given in any unique factorization domain. What about the converse? Can the arithmetic derivative be (well-)defined on a non-unique factorization domain? In the general case, this remains to be seen, but we answer the question negatively for the integers of certain quadratic fields. We also give a sufficient condition under which the answer is negative.
Full version: pdf,
(Concerned with sequences
Received October 30 2012;
revised version received January 1 2013.
Published in Journal of Integer Sequences, January 1 2013.
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