Roger A. Bateman, Elizabeth A. Clark, Michael L. Hancock, and Clifford A. Reiter, The Period of Convergents Modulo M of Reduced Quadratic Irrationals, Fibonacci Quarterly, 29 3 (1991), 220-229.

This paper invesigates the period of the convergent sequences, taken modulo m, of the continued fraction expansions of reduced quadratic irrationals. These sequences generalize the study of such periods for the Fibonacci sequence but have a richer structure. Structure relating to reversals and rotations are noted. Examples of periods for powers of primes equalling smaller powers are given (a counterexample to the analogous conjecture for Fibonacci numbers). Strong divisibility conditions are given when the modulus is prime.

See Also: · A discussion of unusually short periods appears in [nt_1995]. · There are many references to this topic for Fibonacci numbers and variants; one classic reference is: D. D. Wall, Fibonacci Series Modulo m, The American Mathematical Monthly, 67 (1960) 525-32.