Sandy D. Balkin, Deborah S. Cousins, Christopher K. Orr, and Clifford A. Reiter, Short Periods of Continued Fraction Convergents Modulo M: a Generalization of the Fibonacci Case, The Fibonacci Quarterly, 33 3 (1995) 222-233.

This paper investigates reduction formulas that allow the computation of convergents of reduced quadratic irrationals (taken mod m) moving forward by specified factors. When some such formulas are reduced to a canonical form via Simpson type formulas, and one term is adjusted by a power of -1, common factors appear. This has significant impact for the periods appearing when moduli dividing the common factors are considered. Thus, these formula give a method for identifying moduli where the convergents of reduced quadratic irrationals (taken mod m) have unusually short periods.

See Also: · A general discussion of the periods of the convergents of reduced quadratic irrationals taken modulo m appears in [nt_1991]. · The study of the canonical forms of reducton formulas of the type used here, but applied to locating moduli where the Fibonacci sequence has short periods appears in [nt_1993]