Note on a theorem of A. Aiba (Q817252)

Let \(p\) be a rational prime. An integer \(s\in\mathbb Z\) with \(1\leq s\leq p-1\) is said to have property (A) if there exist no integers \(m,n\) with \[ 1\leq n,m\leq p-1\,\text{and}\, \left\{\frac{ns}{p}\right\}<\left\{\frac{ms}{p} \right\} <\frac{s}{p}, \] where for any real \(x\in\mathbb R\) \(\{x\}\in[0,1)\) denotes the fractional part of \(x\). The author notes that for the correctness of the Theorem in [\textit{A. Aiba}, J. Number Theory 102, No. 1, 118--124 (2003; Zbl 1035.11059)] one has to replace the condition in (2)(ii) by (A). For any integer \(n\in\mathbb Z\) let \(r(n)\) denote the least non-negative integer with \(r(n)\equiv n (\text{mod}\,p).\) The author proves: With the above notations, the following statements are equivalent for \(s\geq 2\): (1) \(s\) has property (A). (2) The sequence \((n_1,n_2,\ldots,n_{s-1})\) is strictly monotone decreasing, where \(n_j=r(js^{-1}).\) (3) \(r(s^{-1})>\frac{s-2}{s-1}p\). (4) \(s| (p-1)\). (5) \(r(s^{-1})=p-\frac{p-1}{s}\). As already Aiba mentioned, \(s=1\) obviously has property (A), and also (4) and (5) above hold for \(s=1\).