Connecting Zeros in Pisano Periods to Prime Factors of $K$-Fibonacci Numbers
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arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866913671563706368 |
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| author | Benfield, Brennan Lippard, Oliver |
| author_facet | Benfield, Brennan Lippard, Oliver |
| contents | 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 $m$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_20048 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Connecting Zeros in Pisano Periods to Prime Factors of $K$-Fibonacci Numbers Benfield, Brennan Lippard, Oliver Number Theory 11B39, 11B50 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 $m$. |
| title | Connecting Zeros in Pisano Periods to Prime Factors of $K$-Fibonacci Numbers |
| topic | Number Theory 11B39, 11B50 |
| url | https://arxiv.org/abs/2407.20048 |