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

TL;DR

This paper establishes new mathematical connections between the zeros in Pisano periods and the prime factors of K-Fibonacci numbers, enhancing understanding of their periodic properties and prime factorization relationships.

Contribution

It proves conjectures linking zeros in Pisano periods to prime factors of K-Fibonacci numbers, revealing how index classes determine zero counts.

Findings

Zeros in Pisano periods are exactly 1, 2, or 4, and are evenly spaced.

The distribution of zeros is determined by prime factors of K-Fibonacci numbers.

Congruence classes of indices dictate the number of zeros in the period.

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 $m$.