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<p>We obtain best possible results for the number of coprime positive integer solutions of the equation in the title when <i>a</i> is a positive integer, <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(b=p^{m}\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>b</mi> <mo>=</mo> <msup> <mi>p</mi> <mi>m</mi> </msup> </mrow> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(2p^{m}\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>2</mn> <msup> <mi>p</mi> <mi>m</mi> </msup> </mrow> </math></EquationSource> </InlineEquation> or <InlineEquation ID="IEq5"> <EquationSource Format="TEX">\(4p^{m}\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>4</mn> <msup> <mi>p</mi> <mi>m</mi> </msup> </mrow> </math></EquationSource> </InlineEquation>, where <i>m</i> is a non-negative integer, <i>p</i> is prime, <InlineEquation ID="IEq6"> <EquationSource Format="TEX">\(\gcd \left( a^{2}, b \right) \)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo movablelimits="true">gcd</mo> <mfenced close=")" open="("> <msup> <mi>a</mi> <mn>2</mn> </msup> <mo>,</mo> <mi>b</mi> </mfenced> </mrow> </math></EquationSource> </InlineEquation> is squarefree and <InlineEquation ID="IEq7"> <EquationSource Format="TEX">\(X^{2}- \left( a^{2}+b \right) Y^{2}=-4\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>X</mi> <mn>2</mn> </msup> <mo>-</mo> <mfenced close=")" open="("> <msup> <mi>a</mi> <mn>2</mn> </msup> <mo>+</mo> <mi>b</mi> </mfenced> <msup> <mi>Y</mi> <mn>2</mn> </msup> <mo>=</mo> <mo>-</mo> <mn>4</mn> </mrow> </math></EquationSource> </InlineEquation> has a solution in positive integers. We prove our results by establishing best possible bounds for the number of distinct squares in certain binary recurrence sequences, including those associated with such equations.</p>

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Sharp bounds on the number of squares in recurrence sequences and solutions of \(X^{2}-\left( a^{2}+b \right) Y^{4}=-b\)

  • 期刊论文

  • 发表日期：

    2026年4月4日

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  • Paul M Voutier

摘要

We obtain best possible results for the number of coprime positive integer solutions of the equation in the title when a is a positive integer, \(b=p^{m}\) b = p m , \(2p^{m}\) 2 p m or \(4p^{m}\) 4 p m , where m is a non-negative integer, p is prime, \(\gcd \left( a^{2}, b \right) \) gcd a 2 , b is squarefree and \(X^{2}- \left( a^{2}+b \right) Y^{2}=-4\) X 2 - a 2 + b Y 2 = - 4 has a solution in positive integers. We prove our results by establishing best possible bounds for the number of distinct squares in certain binary recurrence sequences, including those associated with such equations.