Reduced sequences of integers and pseudo-random numbers

where a and bare positive integers. II. The sequence of Fibonacci, defined by u =0, u1=1; u 2=,:i. 1 +u (n=O, 1, •.• ). o n+ n+. n These sequences have the property that if mis a positive integer (where in case I the number mis supposed prime to a) the least non-negative residues mod m of the elements form a periodic sequence. The lengtt, of the period of this sequence will be denoted by C(m). In the'case of I sequence I with a=23, b=47594118 ai.ld m=108+1, Lehmer1) used the least non-negative residues of u0 , ••• ,uc(m)-1 to construct a set of pseudorandom numbers. Our purpose is to investigate properties of the number C(m). In case I we consider primes p with (a,p)=(b,p)=1. Then for p=2 there existsl a positive .. integer k=k(?), such that 2kj aC(4)_1, 2k+1{ aC(4)_1 and for odd p there exists a positiv0 integer k=k(p) such that Pk 1 a C ( P) 1 ' Pk+ 1 { a C ( P) 1 • ( In case I the following relations hold: C(2)=1; C ( p) 1 p-1 t