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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Benfield et al. (2024) studied this question.
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