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arXiv · 2407.20048

Connecting Zeros in Pisano Periods to Prime Factors of $K$-Fibonacci Numbers

Brennan Benfield·Oliver Lippard
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Abstract

The Fibonacci sequence is periodic modulo every positive integer $m>1$, and perhaps more surprisingly, each period has exactly 1, 2, or 4 zeros that are evenly spaced, which also holds true for more general $K$-Fibonacci sequences. This paper proves several conjectures connecting the zeros in the Pisano period to the prime factors of $K$-Fibonacci numbers. The congruence classes of indices for $K$-Fibonacci numbers that are multiples of the prime factors of $m$ completely determine the number of zeroes in the Pisano period modulo $m$.

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Brennan Benfield, Oliver Lippard. 2024-07-29. Connecting Zeros in Pisano Periods to Prime Factors of $K$-Fibonacci Numbers. https://arxiv.org/abs/2407.20048

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