MODULAR MULTIPLICATIVE INVERSES OF FIBONACCI NUMBERS
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초록
Let $F_n, n\in \mathbb{N}$ be the $n-th$ Fibonacci number, and let $(p,q)$ be one of ordered pairs $(F_{n+2},F_n)$ or $(F_{n+1},F_n)$. Then we show that the multiplicative inverse of $q$ mod $p$ as well as that of $p$ mod $q$ are again Fibonacci numbers. For proof of our claim we make use of well-known Cassini, Catlan and d’Ocagne identities. As an application, we determine the number $N_{p,q}$ of nonzero term of a polynomial $\Delta_{p,q}(t)= \frac{(t^{pq}-1)(t-1)}{(t^p-1)(t^q-1)}$ through the Carlitz's formula.
Abstract
Let $F_n, n\in \mathbb{N}$ be the $n-th$ Fibonacci number, and let $(p,q)$ be one of ordered pairs $(F_{n+2},F_n)$ or $(F_{n+1},F_n)$. Then we show that the multiplicative inverse of $q$ mod $p$ as well as that of $p$ mod $q$ are again Fibonacci numbers. For proof of our claim we make use of well-known Cassini, Catlan and d’Ocagne identities. As an application, we determine the number $N_{p,q}$ of nonzero term of a polynomial $\Delta_{p,q}(t)= \frac{(t^{pq}-1)(t-1)}{(t^p-1)(t^q-1)}$ through the Carlitz's formula.
참고문헌 (5)
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