A common pseudorandom number generator is the power generator: x ↦ → x ℓ (mod n). Here, ℓ, n are fixed integers at least 2, and one constructs a pseudorandom sequence by starting at some residue mod n and iterating this ℓth power map. (Because it is the easiest to compute, one often takes ℓ = 2; this case is known as the BBS generator, for Blum,

We use character sums to confirm several recent conjectures of V. I. Arnold on the uniformity of distribution properties of a certain dynamical system in a finite field. On the other hand, we show that some conjectures are wrong. We also analyze several other conjectures of V. I. Arnold related to the orbit length of similar dynamical systems in residue rings and outline possible ways to prove them. We also show that some of them require further tuning. 1

Introduced by Kraitchik and Lehmer, an x-pseudosquare is a positive integer n ≡ 1 (mod 8) that is a quadratic residue for each odd prime p ≤ x, yet is not a square. We give a subexponential upper bound for the least x-pseudosquare that improves on a bound that is exponential in x due to Schinzel. We also obtain an equi-distribution result for pseudosquares. An x-pseudopower to base g is a positive integer which is not a power of g yet is so modulo p for all primes p ≤ x. It is conjectured by Bach, Lukes, Shallit, and Williams that the least such number is at most exp(agx/log x) for a suitable constant ag. A bound of exp(agxlog log x/log x) is proved conditionally on the Riemann Hypothesis for Dedekind zeta functions, thus improving on a recent conditional exponential bound of Konyagin and the present authors. We also give a GRH-conditional equidistribution result for pseudopowers that is analogous to our unconditional result for pseudosquares.