Journal of the Korean Mathematical Society (대한수학회지)

Korean Mathematical Society (대한수학회)
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SOME CONDITIONS ON THE FORM OF THIRD ELEMENT FROM DIOPHANTINE PAIRS AND ITS APPLICATION

  • Lee, June Bok (Department of Mathematical Sciences Yonsei University) ;
  • Park, Jinseo (Department of Mathematics Education Catholic Kwandong University)
  • Received : 2017.04.29
  • Accepted : 2017.06.19
  • Published : 2018.03.01
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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.

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References

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