The p-adic order of some Fibonomial coefficients

Result code in IS VaVaI
<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F62690094%3A18470%2F15%3A50003461" target="_blank" >RIV/62690094:18470/15:50003461 - isvavai.cz</a>
Result on the web
<a href="https://cs.uwaterloo.ca/journals/JIS/VOL18/Marques2/marques11.pdf" target="_blank" >https://cs.uwaterloo.ca/journals/JIS/VOL18/Marques2/marques11.pdf</a>
DOI - Digital Object Identifier
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Result language
angličtina
Original language name
The p-adic order of some Fibonomial coefficients
Original language description
Let $(F_n)_{ngeq 0}$ be the Fibonacci sequence. For $1le kle m$, the Fibonomial coefficient is defined as $$atopwithdelims{m}{k}= frac{F_{m-k+1}cdots F_{m-1} F_{m}}{F_1cdots F_k}.$$ In 2013, the authors and Sellers proved that if $p$ is a prime number such that $pequiv -2$ or $2pmod 5$, then $p mid atopwithdelims{p^{a+1}}{ p^a}$ for all integers $ageq 1$. In this paper, we generalize this result by proving the exact division [ p^{lceil (a+delta_{p,2})/2rceil} parallel atopwithdelims{p^{a+1}}{ p^a} ,] for all prime $p$ and $ageq 1$ (where $delta_{i,j}$ denotes the Kronecker delta). Moreover, we shall prove that if $pequiv -1$ or $1pmod 5$, then $p nmid atopwithdelims{p^{a+1}}{ p^a}$ for all integers $ageq 1$ confirming therefore a recent conjecture.
Czech name
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Czech description
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Type
J<sub>x</sub> - Unclassified - Peer-reviewed scientific article (Jimp, Jsc and Jost)
CEP classification
BA - General mathematics
OECD FORD branch
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Project
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Continuities
I - Institucionalni podpora na dlouhodoby koncepcni rozvoj vyzkumne organizace

Publication year
2015
Confidentiality
S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů

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