On a Congruence Involving Generalized Fibonomial Coefficients

Result code in IS VaVaI
<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F62690094%3A18470%2F18%3A50014382" target="_blank" >RIV/62690094:18470/18:50014382 - isvavai.cz</a>
Result on the web
<a href="http://dx.doi.org/10.1134/S2070046618010053" target="_blank" >http://dx.doi.org/10.1134/S2070046618010053</a>
DOI - Digital Object Identifier
<a href="http://dx.doi.org/10.1134/S2070046618010053" target="_blank" >10.1134/S2070046618010053</a>

Result language
angličtina
Original language name
On a Congruence Involving Generalized Fibonomial Coefficients
Original language description
Let $(F_n)_{ngeq 0}$ be the Fibonacci sequence. For $1le kle m$, the Fibonomial coefficient is defined as atopwithdelims{n}{k}= frac{F_{n-k+1}cdots F_{n-1} F_{n}}{F_1cdots F_k}. In 2013, Marques, Sellers and Trojovsk' y proved that if $p$ is a prime number such that $pequiv pm 1 pmod 5$, then $p nmid atopwithdelims{p^{a+1}}{p^a}$ for all integers $ageq 1$. In 2010, in particular, Kilic generalized the Fibonomial coefficients for fibm{n}{k}= frac{F_{(n-k+1)m}cdots F_{(n-1)m} F_{nm}}{F_mcdots F_{km}}. In this paper, we generalize Marques, Sellers and Trojovsky result to prove, in particular, that if $pequiv pm 1pmod 5$, then $fibm{p^{a+1}}{p^a}equiv 1pmod{p}$, for all $ageq 0$ and $mgeq 1$.
Czech name
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Czech description
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Type
J<sub>SC</sub> - Article in a specialist periodical, which is included in the SCOPUS database
CEP classification
—
OECD FORD branch
10101 - Pure mathematics

Publication year
2018
Confidentiality
S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů

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