Let D be a positive integer and p an odd prime with <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11139_2025_1061_Article_IEq1.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>. Assume that the equation <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11139_2025_1061_Article_IEq2.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="112" /> </InlineMediaObject> <EquationSource Format="TEX">$u^2-Dv^2=-1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>u</mi> <mn>2</mn> </msup> <mo>-</mo> <mi>D</mi> <msup> <mi>v</mi> <mn>2</mn> </msup> <mo>=</mo> <mo>-</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation> has integer solutions (u, v) and that the fundamental solution <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11139_2025_1061_Article_IEq3.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="80" /> </InlineMediaObject> <EquationSource Format="TEX">$U_1+V_1\sqrt{p}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>U</mi> <mn>1</mn> </msub> <mo>+</mo> <msub> <mi>V</mi> <mn>1</mn> </msub> <msqrt> <mi>p</mi> </msqrt> </mrow> </math></EquationSource> </InlineEquation> to the Pell equation <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11139_2025_1061_Article_IEq4.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="101" /> </InlineMediaObject> <EquationSource Format="TEX">$U^2-pV^2=1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>U</mi> <mn>2</mn> </msup> <mo>-</mo> <mi>p</mi> <msup> <mi>V</mi> <mn>2</mn> </msup> <mo>=</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation> satisfies <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11139_2025_1061_Article_IEq5.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="41" /> </InlineMediaObject> <EquationSource Format="TEX">$p \not \mid V_1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>p</mi> <mo>∤</mo> <msub> <mi>V</mi> <mn>1</mn> </msub> </mrow> </math></EquationSource> </InlineEquation>. In this paper, we then prove that the number of positive integer solutions (x, n) to the generalized Ramanujan–Nagell equation <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11139_2025_1061_Article_IEq6.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="92" /> </InlineMediaObject> <EquationSource Format="TEX">$x^2-D=p^n$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>x</mi> <mn>2</mn> </msup> <mo>-</mo> <mi>D</mi> <mo>=</mo> <msup> <mi>p</mi> <mi>n</mi> </msup> </mrow> </math></EquationSource> </InlineEquation> is at most two.

On the number of solutions to the generalized Ramanujan–Nagell equation

期刊论文
发表日期：

2025年3月13日

查看全文

Yasutsugu Fujita,
Maohua Le

摘要

Let D be a positive integer and p an odd prime with $p \not \mid D$ $p ∤ D$ . Assume that the equation $u^2-Dv^2=-1$ $u^{2} - D v^{2} = - 1$ has integer solutions (u, v) and that the fundamental solution $U_1+V_1\sqrt{p}$ $U_{1} + V_{1} \sqrt{p}$ to the Pell equation $U^2-pV^2=1$ $U^{2} - p V^{2} = 1$ satisfies $p \not \mid V_1$ $p ∤ V_{1}$ . In this paper, we then prove that the number of positive integer solutions (x, n) to the generalized Ramanujan–Nagell equation $x^2-D=p^n$ $x^{2} - D = p^{n}$ is at most two.