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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.
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참고문헌 (15)
참고문헌 신청
A. Baker / 1969 / The equations $3^x-2=y^2$ and $8^x-7=z^2$ / Quart. J. Math. Oxford Ser. (2) 20:129~137
A. Baker / 1993 / Logarithmic forms and group varieties / J. Reine Angew. Math. 442:19~62
Y. Bugeaud / 2007 / On the family of Diophantine triples {k-1+1, $16k^3$ -4k} / Glasg. Math. J. 49(2):333~344
A. Dujella / 1997 / The problem of the extension of a parametric family of Diophantine triples / Publ. Math. Debrecen 51(3-4):311~322
A. Dujella / 2000 / A parametric family of elliptic curves / Acta Arith. 94(1):87~101
A. Baker / 1993 / Logarithmic forms and group varieties / J. Reine Angew. Math. 442:19~62
Y. Bugeaud / 2007 / On the family of Diophantine triples {k-1+1,
A. Dujella / 1997 / The problem of the extension of a parametric family of Diophantine triples / Publ. Math. Debrecen 51(3-4):311~322
A. Dujella / 2000 / A parametric family of elliptic curves / Acta Arith. 94(1):87~101
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