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The Period, Rank, and Order of the (a, b)-Fibonacci Sequence Mod m
Vol. 86, No. 5 (December 2013), pp. 372-380
Published by: Mathematical Association of America
Stable URL: http://www.jstor.org/stable/10.4169/math.mag.86.5.372
Page Count: 9
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Summary For given integers a and b, we consider the (a, b)-Fibonacci sequence F defined by F0 = 0, F1 = 1, and Fn = aFn−1 + bFn−2. Given m ≥ 2 relatively prime to b, F (mod m) is periodic with period denoted π(m). The rank of F (mod m), denoted α(m), is the least positive r such that Fr ≣ 0 (mod m), and the order of F (mod m), denoted ω(m), is π(m)/α(m). In this article, we pull together results on π(m), α(m), and ω(m) from the classic case a = 1, b = 1, and generalize their proofs to accommodate arbitrary integers a and b. Matrix methods are used extensively to provide elementary proofs.
Copyright the Mathematical Association of America 2013