Jacobi symbols and Eulerʼs number e

Let pk/qkpk/qk, k⩾0k⩾0, be the convergents of the continued fraction expansion of a number x∈R∖Qx∈R∖Q. We investigate the sequence of Jacobi symbols (pkqk), k⩾0k⩾0. We show that this sequence is purely periodic with period length 24 for x=e=2.718281⋯x=e=2.718281⋯ and period length 40 for x=e2x=e2. Further, we take the first steps towards a general theory of such sequences of Jacobi symbols. For instance, we show that there are uncountably many numbers x such that this sequence has period 1, and that every natural number L actually occurs as the period length of some x.