For binary recurrence sequences, <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\left( y_{k} \right) _{k \in \mathbb {Z}}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mfenced close=")" open="("> <msub> <mi>y</mi> <mi>k</mi> </msub> </mfenced> <mrow> <mi>k</mi> <mo>∈</mo> <mi mathvariant="double-struck">Z</mi> </mrow> </msub> </math></EquationSource> </InlineEquation>, arising from the solutions of generalised Pell equations, <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$X^{2}-dY^{2}=c$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>X</mi> <mn>2</mn> </msup> <mo>-</mo> <mi>d</mi> <msup> <mi>Y</mi> <mn>2</mn> </msup> <mo>=</mo> <mi>c</mi> </mrow> </math></EquationSource> </InlineEquation>, where <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$y_{0}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>y</mi> <mn>0</mn> </msub> </math></EquationSource> </InlineEquation> is any positive square and <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$c=-2^{\ell }p^{m}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>c</mi> <mo>=</mo> <mo>-</mo> <msup> <mn>2</mn> <mi>ℓ</mi> </msup> <msup> <mi>p</mi> <mi>m</mi> </msup> </mrow> </math></EquationSource> </InlineEquation> for an odd prime, p, and non-negative integers <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$\ell $</EquationSource> <EquationSource Format="MATHML"><math> <mi>ℓ</mi> </math></EquationSource> </InlineEquation> and m, we show that there are at most 4 distinct squares with <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$y_{k}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>y</mi> <mi>k</mi> </msub> </math></EquationSource> </InlineEquation> sufficiently large. From this result, we also show that there are at most 7 distinct squares when <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$y_{0}=1,2^{2},\ldots ,7^{2}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>y</mi> <mn>0</mn> </msub> <mo>=</mo> <mn>1</mn> <mo>,</mo> <msup> <mn>2</mn> <mn>2</mn> </msup> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msup> <mn>7</mn> <mn>2</mn> </msup> </mrow> </math></EquationSource> </InlineEquation>, or once d exceeds an explicit lower bound, without any conditions on the size of such squares.

Bounds on the number of squares in recurrence sequences: $y_{0}=b^{2}$ (II)

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发表日期：

2026年1月12日

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Paul M Voutier

摘要

For binary recurrence sequences, $\left( y_{k} \right) _{k \in \mathbb {Z}}$ ${(y_{k})}_{k \in Z}$ , arising from the solutions of generalised Pell equations, $X^{2}-dY^{2}=c$ $X^{2} - d Y^{2} = c$ , where $y_{0}$ $y_{0}$ is any positive square and $c=-2^{\ell }p^{m}$ $c = - 2^{ℓ} p^{m}$ for an odd prime, p, and non-negative integers $\ell $ $ℓ$ and m, we show that there are at most 4 distinct squares with $y_{k}$ $y_{k}$ sufficiently large. From this result, we also show that there are at most 7 distinct squares when $y_{0}=1,2^{2},\ldots ,7^{2}$ $y_{0} = 1, 2^{2}, \dots, 7^{2}$ , or once d exceeds an explicit lower bound, without any conditions on the size of such squares.

Bounds on the number of squares in recurrence sequences: \(y_{0}=b^{2}\) (II)

摘要