On Diophantine equations involving Lucas sequences

Result code in IS VaVaI
<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F62690094%3A18470%2F19%3A50015685" target="_blank" >RIV/62690094:18470/19:50015685 - isvavai.cz</a>
Result on the web
<a href="https://www.degruyter.com/view/j/math.2019.17.issue-1/math-2019-0073/math-2019-0073.xml" target="_blank" >https://www.degruyter.com/view/j/math.2019.17.issue-1/math-2019-0073/math-2019-0073.xml</a>
DOI - Digital Object Identifier
<a href="http://dx.doi.org/10.1515/math-2019-0073" target="_blank" >10.1515/math-2019-0073</a>

Result language
angličtina
Original language name
On Diophantine equations involving Lucas sequences
Original language description
In this paper, we study the Diophantine equation $u_n=R(m)P(m)^{Q(m)}$,where $R, P$ and $Q$ are some polynomials (under weak assumptions) and $u_n$ is a Lucas sequence, thus the sequence $(u_n)_{ngeq 0}$ with characteristic polynomial $f(x) = x^2-ax-b$, i.e., $(u_n)_{ngeq 0}$ is the integral sequence satisfying $u_0=0, u_1=1$, and $u_n = au_{n-1} +bu_{n-2}$, for all integers $ngeq 2$. We suppose that this sequence is non degenerated.In this paper, we describe how a method based on $p$-adic valuations can be settled to this kind of equation. We found a upper bound for solutions of special case of this Diophantine equation in the form $F_n=km^m(m+1)$, where $k,m,n$ are any given positive integer.
Czech name
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Czech description
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Type
J<sub>imp</sub> - Article in a specialist periodical, which is included in the Web of Science database
CEP classification
—
OECD FORD branch
10101 - Pure mathematics

Publication year
2019
Confidentiality
S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů

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