Quadratic congruential pseudorandom numbers: distribution of triples

The present paper deals with the quadratic congruential method with power of two modulus for generating uniform pseudorandom numbers. Statistical independence properties of pairs and triples of successive terms in the generated sequences are studied based on the discrepancy of the corresponding point sets. In the main part of the paper a known upper bound for the discrepancy of pairs is improved and new upper bounds for the discrepancy of triples are established. Known lower bounds for the corresponding discrepancies are recalled. The method of proof relies on a detailed discussion of the properties of certain exponential sums.