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Let \(\{\mathcal {H}_{r}\}_{r\ge 0}\) be a sequence of the first kind of Lucas, defined by the recurrence relation \(\mathcal {H}_{r}=p\mathcal {H}_{r-1}-q\mathcal {H}_{r-2}\) for all \(r\ge 2\) , where \(p\ge 1\) and \(q\in \{-1,1\}\) . The initial conditions are \(\mathcal {H}_{0}=0\) and \(\mathcal {H}_{1}=1\) . The first and second kinds of Thabit numbers are given by \(3\cdot 2^{l}-1\) and \(3\cdot 2^{l}+1\) for a non-negative integer l, respectively. In this paper, we have discovered effective bounds for the solutions of the Diophantine equations \(\mathcal {H}_{r}\pm \mathcal {H}_{s}=3\cdot 2^{l}\mp 1\) , in non-negative integers r, s, l, and \(\mathcal {H}_{r}\cdot \mathcal {H}_{s}=3\cdot 2^{l}\mp 1\) , in positive integers r, s with \(l\ge 0\) . In particular, we solve these Diophantine equations for cases in which \((p,q)=\{(2,-1),(1,-1)\}\) .

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The First and Second Kinds of Thabit Numbers Expressed as Sums, Differences or Products of Two Generalized Lucas Sequences

  • 会议论文

  • 发表日期：

    2026年

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  • Hunar Sherzad Taher,
  • Saroj Kumar Dash

摘要

Let \(\{\mathcal {H}_{r}\}_{r\ge 0}\) be a sequence of the first kind of Lucas, defined by the recurrence relation \(\mathcal {H}_{r}=p\mathcal {H}_{r-1}-q\mathcal {H}_{r-2}\) for all \(r\ge 2\) , where \(p\ge 1\) and \(q\in \{-1,1\}\) . The initial conditions are \(\mathcal {H}_{0}=0\) and \(\mathcal {H}_{1}=1\) . The first and second kinds of Thabit numbers are given by \(3\cdot 2^{l}-1\) and \(3\cdot 2^{l}+1\) for a non-negative integer l, respectively. In this paper, we have discovered effective bounds for the solutions of the Diophantine equations \(\mathcal {H}_{r}\pm \mathcal {H}_{s}=3\cdot 2^{l}\mp 1\) , in non-negative integers r, s, l, and \(\mathcal {H}_{r}\cdot \mathcal {H}_{s}=3\cdot 2^{l}\mp 1\) , in positive integers r, s with \(l\ge 0\) . In particular, we solve these Diophantine equations for cases in which \((p,q)=\{(2,-1),(1,-1)\}\) .