Some Properties of the Fibonacci Sequence

The purposes of this paper are; (a) to develop a relationship between subscripts of the symbols of Fibonacci and Lucas numbers and the numbers themselves; (b) to develop relationships between selected modulo and the lengths of the periods of the non-negative residue classes of several general Fibonacci sequences; and (c) to show, through the use of continued fractions, that certain Diophantine equations have solutions in Fibonacci numbers. General terms for the Fibonacci and Lucas sequences are developed through the use of differential equations. These general terms are then used to show that any Fibonacci or Lucas number is a function of its subscript. The length of the periods of the least non-negative residue classes for a given modulus is considered and is found to be the same for any of the general Fibonacci sequences. The exception is the Lucas sequence modulo five. The length of its period is a multiple of the selected general Fibonacci sequences modulo five. It is further shown that a Diophantine equation of the form fx ± ty = c where f and t are either consecutive Fibonacci or consecutive Lucas numbers, has a solution in Fibonacci numbers.