On the Period Length Modulo D of Sequences of Numerators and Denominators of Convergents for the Square Root of a Non-square D

In this paper we investigate the properties of sequences of numerators and denominators of convergents for \(\sqrt{D},\) where D is not a perfect square. In our previous work we prove that \(L = l, 2l\) or 4l, where L is the period length modulo p of the numerators of convergents for \(\sqrt{p}\) and l is the period length of the continued fraction for \(\sqrt{p},\) where p is a prime. Namely, if \(p=2\) or \(p\equiv 7\pmod 8,\) then \(L=l;\) if \(p\equiv 3\pmod 8,\) then \(L=2l;\) if \(p\equiv 1\pmod 4,\) then \(L=4l\) . Here we generalize this result and prove that \(L=l,2l\) or 4l for convergents for \(\sqrt{D},\) where D is not a perfect square. Moreover, we prove that the shortest period length \(L_B\) of the sequence of denominators of convergents for \(\sqrt{D}\) modulo D is equal to \(\frac{D}{\gcd {(D,B_{L-1})}} \cdot L.\) In addition, we prove under the assumption of Ankeny-Artin-Chowla and Mordell conjectures that \(L_B=pL\) for a prime p. We also discuss a possible application of our results to image processing and encoding using a discrete version of the Arnold’s cat map.