McKenzie, Harvey C.||Beach, James W.
M.S. (Master of Science)
Department of Mathematics
Fibonacci numbers||Number theory
This paper treats the cycles formed by the residues of different Fibonacci series under prime modulis. I begin with a few elementary definitions and some background theorems which are assumed true. The basic relation under study is the comparison of lengths of different series and methods of predicting when cycles of shorter length than others of the same moduli occur. I first develop some foundational theorems true for all moduli and then consider two methods of determining short cycles. The first method, and its accompanying theorems, depends upon the use of exponential cycles under prime moduli. The bases for these cycles are solutions to the congruence a² ≡ a + 1 (mod p). The second method rests upon the concept of inverse cycles. These are defined and methods established for determining their existence. I conclude by showing the relationship between these two approaches and by summarizing the main topics of the paper.
Fischer, Roger, "Cycles formed from fibonacci series under prime moduli" (1965). Graduate Research Theses & Dissertations. 2034.
ii, 24 pages
Northern Illinois University
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