Some conditions on the form of third element from Diophantine pairs and its application
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A set $\{a_1, a_2, \ldots, a_m\}$ of positive integers is called a Diophantine $m$-tuple if $a_ia_j+1$ is a perfect square for all $1\leq i < j \leq m$. In this paper, we show that the form of third element in Diophantine pairs and develop some results which are needed to prove the extendibility of the Diophantine pair $\{a, b\}$ with some conditions. By using this result, we prove the extendibility of Diophantine pairs $\{F_{k-2}F_{k+1}, F_{k-1}F_{k+2}\}$ and $\{F_{k-2}F_{k-1},$ $F_{k+1}F_{k+2}\}$, where $F_n$ is the $n$-th Fibonacci number.
Abstract
A set $\{a_1, a_2, \ldots, a_m\}$ of positive integers is called a Diophantine $m$-tuple if $a_ia_j+1$ is a perfect square for all $1\leq i < j \leq m$. In this paper, we show that the form of third element in Diophantine pairs and develop some results which are needed to prove the extendibility of the Diophantine pair $\{a, b\}$ with some conditions. By using this result, we prove the extendibility of Diophantine pairs $\{F_{k-2}F_{k+1}, F_{k-1}F_{k+2}\}$ and $\{F_{k-2}F_{k-1},$ $F_{k+1}F_{k+2}\}$, where $F_n$ is the $n$-th Fibonacci number.
참고문헌 (15)
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