Let D be an odd positive integer, and let p be an odd prime with <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="605_2025_2062_Article_IEq3.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="42" /> </InlineMediaObject> <EquationSource Format="TEX">$p \not \mid D$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>p</mi> <mo>∤</mo> <mi>D</mi> </mrow> </math></EquationSource> </InlineEquation>. Denote by N(D, p) the number of positive integer solutions (x, n) to the generalized Ramanujan–Nagell equation <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="605_2025_2062_Article_IEq4.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="100" /> </InlineMediaObject> <EquationSource Format="TEX">$x^2-D=4p^n$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>x</mi> <mn>2</mn> </msup> <mo>-</mo> <mi>D</mi> <mo>=</mo> <mn>4</mn> <msup> <mi>p</mi> <mi>n</mi> </msup> </mrow> </math></EquationSource> </InlineEquation>. In this paper, we prove <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="605_2025_2062_Article_IEq5.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="91" /> </InlineMediaObject> <EquationSource Format="TEX">$N(D,p)\le 3$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>N</mi> <mo stretchy="false">(</mo> <mi>D</mi> <mo>,</mo> <mi>p</mi> <mo stretchy="false">)</mo> <mo>≤</mo> <mn>3</mn> </mrow> </math></EquationSource> </InlineEquation>.

On the number of solutions to the generalized Ramanujan–Nagell equation $x^2-D=4p^n$

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发表日期：

2025年2月19日

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Yasutsugu Fujita,
Maohua Le

摘要

Let D be an odd positive integer, and let p be an odd prime with $p \not \mid D$ $p ∤ D$ . Denote by N(D, p) the number of positive integer solutions (x, n) to the generalized Ramanujan–Nagell equation $x^2-D=4p^n$ $x^{2} - D = 4 p^{n}$ . In this paper, we prove $N(D,p)\le 3$ $N (D, p) \leq 3$ .

On the number of solutions to the generalized Ramanujan–Nagell equation \(x^2-D=4p^n\)

摘要