We give a simple condition for a linear recurrence (mod 2 w ) of degree r to have the maximal possible period 2 w 1 (2 r 1). It follows that the period is maximal in the cases of interest for pseudo-random number generation, i.e. for 3-term linear recurrences dened by trinomials which are primitive (mod 2) and of degree r > 2. We consider the enumeration of certain exceptional polynomials which do not give maximal period, and list all such polynomials of degree less than 15. 1.

We consider the requirements for uniform pseudo-random number generators on modern vector and parallel supercomputers, consider the pros and cons of various classes of methods, and outline what is currently available. We propose a class of random number generators which have good statistical properties and can be implemented efficiently on vector processors and parallel machines. A good method for initialization of these generators is described, and an implementation on a Fujitsu VP 2200/10 vector processor is discussed. 1

We consider the requirements for uniform pseudo-random number generators on modern vector and parallel machines; consider the pros and cons of various popular classes of methods and some new methods; and outline what is currently available. We then make a proposal for a class of random number generators which have good statistical properties and can be implemented efficiently on vector processors and parallel machines. A proposal regarding initialization of these generators is made. We also discuss the results of a trial implementation on a Fujitsu VP 2200/10 vector processor.

Any one who considers arithmetical methods of produc ing random digits is, of course, in a state of sin. " These wry words were written 40 years ago by John von Neu mann, the Hungarian-American mathematician and pio neer of computing. He went on to explain: "As has been pointed out several times, there is no such thing as a ran dom number—there are only methods to produce random numbers, and a strict arithmetic procedure of course is not such a method." Von Neumann's argument would seem to rule out any possibility of having a computer generate random numbers. A properly functioning digital computer must at all mo ments follow an algorithm—von Neumann's "strict arith metic procedure"—which means the computer's actions are entirely deterministic. The path taken by the algorithm may