Basic Properties of Primitive Root and Order Function

The notation and terminology used here have been introduced in the following papers: [1], [18], [9], [4], [7], [5], [20], [16], [17], [19], [14], [2], [15], [3], [10], [13], [22], [23], [21], and [6]. For simplicity, we adopt the following convention: i, s, t, m, n, k are natural numbers, d, e are elements of N, f1 is a finite sequence of elements of N, and x is an integer. Let m be a natural number. The functor RelPrimesm yields a set and is defined by: (Def. 1) RelPrimesm = {k ∈ N: m and k are relative prime ∧ 1 ≤ k ∧ k ≤ m}. We now state the proposition (1) RelPrimesm ⊆ Segm. Let m be a natural number. Then RelPrimesm is a subset of N. Let m be a natural number. Observe that RelPrimesm is finite. Next we state several propositions: (2) If 1 ≤ m, then RelPrimesm 6= ∅. Authors thank Andrzej Trybulec and Yatsuka Nakamura for the help during writing this article.