On decompositions of quadrinomials and related Diophantine equations

Maciej Gawron¹

¹ Institute of Mathematics Jagiellonian University Łojasiewicza 6 30-348 Kraków, Poland

Acta Arithmetica, Tome 179 (2017) no. 1, pp. 1-15

Résumé

Let $A,B,C,D$ be non-zero rational numbers, and let $n_1,n_2,n_3$ be distinct positive integers. We solve the equation \begin{equation*} Ax^{n_1}+Bx^{n_2}+Cx^{n_3}+D = f(g(x)) \end{equation*} in $f,g \in \mathbb{Q}[x]$. Then we use the Bilu–Tichy method to prove that the equation \begin{equation*} Ax^{n_1}+Bx^{n_2}+Cx^{n_3}+D = Ey^{m_1}+Fy^{m_2}+Gy^{m_3}+H \end{equation*} has finitely many integral solutions where $A,B,C,D,E,F,G,H$ are non-zero rational numbers and $(n_1,n_2,n_3)$, $(m_1,m_2,m_3)$ are different triples of distinct positive integers such that $\gcd(n_1,n_2,n_3) = \gcd(m_1,m_2,m_3)=1$ and $n_1,m_1 \geq 9$. We establish the same result for the equation \begin{equation*} A_1x^{n_1}+A_2x^{n_2}+\cdots+A_l x^{n_l} + A_{l+1} = Ey^{m_1}+Fy^{m_2}+Gy^{m_3}, \end{equation*} where $l \geq 4$ is a fixed integer, $A_1,\ldots,A_{l+1},E,F,G$ are rational numbers, non-zero except possibly for $A_{l+1}$, and $n_1,\ldots,n_l$ and $m_1,m_2,m_3$ are sequences of distinct positive integers such that $\gcd(n_1, \ldots n_l) = \gcd(m_1,m_2,m_3)=1$ and $n_1 \gt 2l$, $m_1 \geq 2l(l-1)$.

Détail

DOI : 10.4064/aa8411-9-2016

Keywords: d non zero rational numbers distinct positive integers solve equation begin equation* end equation* mathbb bilu tichy method prove equation begin equation* end equation* has finitely many integral solutions where d h non zero rational numbers different triples distinct positive integers gcd gcd geq establish result equation begin equation* cdots end equation* where geq fixed integer ldots rational numbers non zero except possibly ldots sequences distinct positive integers gcd ldots gcd geq l