  For any positive integer $q$, the sequence of the Euler up/down numbers
reduced modulo $q$ was proved to be ultimately periodic by Knuth and Buckholtz.
Based on computer simulations, we state for each value of $q$ precise
conjectures for the minimal period and for the position at which the sequence
starts being periodic. When $q$ is a power of $2$, a sequence defined by Arnold
appears, and we formulate a conjecture for a simple computation of this
sequence.
