Two classes of piecewise-linear difference equations with eventual periodicity three (Q882038)

The author proves two of the conjectures stated in his previous paper [ibid. 295, No. 2, 570--575 (2004; Zbl 1058.39002)]. More precisely, he proves the following two theorems. Let \(\alpha\) be an integer greater than 2 and let \(\{x_n\}\) be a nontrivial solution of the equation \(x_{n+1}=(Ax)_n\), where \((Ax)_n=(x_n+x_{n-1})/\alpha\) if \(\alpha | x_n+x_{n-1}\) and \((Ax)_n=-x_n-x_{n-1}\) otherwise. Then \(\{x_n\}\) is eventually periodic of prime period 3. Let \(\{x_n\}\) be a nontrivial solution of the equation \(x_{n+1}=(Bx)_n\), where \((Bx)_n=(x_n-x_{n-1})/3\) if \(3 | x_n-x_{n-1}\) and \((Bx)_n=-x_n-x_{n-1}\) otherwise. Then \(\{x_n\}\) is eventually periodic of prime period 3.