Arithmetical Observation

1. Let a and b denote positive integers that are relatively prime, and let r1, r2, r3, ... be the non-negative least residues of the numbers a, 2a, 3a,... modulo b, which we assume to be greater than the number a.
This series of residues

arises from the indefinite repetition of the period

which itself, ignoring the order, corresponds to the numbers 0, 1, 2, ... b — 1, with the first residue being r1 = a, and the last being 0.

Now, in a very different investigation, I recently happened to make an observation which seems to illustrate singularly the arithmetical nature of such a series of residues.

The question at hand is when rm increases or decreases if m is increased by unity. To decide this question, where the series (r.) itself is available, nothing need be done except to observe whether the residue rm is succeeded by a larger or smaller number rm+1.

2. That which is observed in such a way, by examining the series (r.) regarding the increasing or decreasing values of rm, is expressed excellently for our purpose if it is denoted by a continuous series of the letters c or d, according to whether rm increases or decreases, i.e., according to whether a larger or smaller number rm+1 is found after rm.

In this way a new series is born, composed of only two letters, c and d, but in a certain order: the terms of which, if it is necessary to consider the position they occupy in that series, we denote by