Loop Properties and Controllability of Linear Congruential Sequences

Loop properties of linear congruential sequences generated by Xn+1 = AXn + C (mod m) are considered. Existence theorems and procedures are given to single out loops and fixed points, if any, in terms of generator parameters X0,A,C,m. Criteria are found for absolutely and conditionally convergent generators and split generators. The role of parameters in respect of inner structure is investigated. Nonautonomous random number generators are introduced and applied to the design of a generator (mod 232) having a good lattice, a period five times more than the modulus and no translated-block structure.