Clifford A. Reiter, Fibonacci Numbers: Reduction Formulas and Short Periods, The Fibonacci Quarterly, 31 4 (1993) 315-324.

This paper investigates reduction formulas that allow the computation of Fibonacci pairs moving forward by specified factors. When some such formulas are reduced to a canonical form via Simpson formulas, and one term is adjusted by a power of -1, common factors appear. In some cases this facilitates computations, but it 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 Fibonacci sequence has unusually short periods.

See Also: · While the application to these periods is different, the reduction formulas of the type studied here were useful for rapid computation of very large Fibonacci numbers [nt_1992]. · These reduction formulas and their application to identifying moduli giving rise to unusually short periods were generalized to reduced quadratic irrationals in [nt_1995].