Let <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$D_1$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>D</mi> <mn>1</mn> </msub> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$D_2$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>D</mi> <mn>2</mn> </msub> </math></EquationSource> </InlineEquation> be positive integers and let p be an odd prime number such that <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$D:=D_1D_2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>D</mi> <mo>:</mo> <mo>=</mo> <msub> <mi>D</mi> <mn>1</mn> </msub> <msub> <mi>D</mi> <mn>2</mn> </msub> </mrow> </math></EquationSource> </InlineEquation> is nonsquare and <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\gcd (D,2p)=1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo movablelimits="true">gcd</mo> <mo stretchy="false">(</mo> <mi>D</mi> <mo>,</mo> <mn>2</mn> <mi>p</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>. Denote by <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$N(D_1,D_2,4,p)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>N</mi> <mo stretchy="false">(</mo> <msub> <mi>D</mi> <mn>1</mn> </msub> <mo>,</mo> <msub> <mi>D</mi> <mn>2</mn> </msub> <mo>,</mo> <mn>4</mn> <mo>,</mo> <mi>p</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> the number of positive integer solutions (x, n) to the generalized Ramanujan–Nagell equation <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$D_1x^2-D_2=4p^n$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>D</mi> <mn>1</mn> </msub> <msup> <mi>x</mi> <mn>2</mn> </msup> <mo>-</mo> <msub> <mi>D</mi> <mn>2</mn> </msub> <mo>=</mo> <mn>4</mn> <msup> <mi>p</mi> <mi>n</mi> </msup> </mrow> </math></EquationSource> </InlineEquation>. In this paper, we prove <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$N(D_1,D_2,4,p)\le 4$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>N</mi> <mo stretchy="false">(</mo> <msub> <mi>D</mi> <mn>1</mn> </msub> <mo>,</mo> <msub> <mi>D</mi> <mn>2</mn> </msub> <mo>,</mo> <mn>4</mn> <mo>,</mo> <mi>p</mi> <mo stretchy="false">)</mo> <mo>≤</mo> <mn>4</mn> </mrow> </math></EquationSource> </InlineEquation>. Further, we prove that if the squarefree part of <InlineEquation ID="IEq10"> <EquationSource Format="TEX">$D_1$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>D</mi> <mn>1</mn> </msub> </math></EquationSource> </InlineEquation> is at most 1001, then <InlineEquation ID="IEq11"> <EquationSource Format="TEX">$N(D_1,D_2,4,p)\le 3$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>N</mi> <mo stretchy="false">(</mo> <msub> <mi>D</mi> <mn>1</mn> </msub> <mo>,</mo> <msub> <mi>D</mi> <mn>2</mn> </msub> <mo>,</mo> <mn>4</mn> <mo>,</mo> <mi>p</mi> <mo stretchy="false">)</mo> <mo>≤</mo> <mn>3</mn> </mrow> </math></EquationSource> </InlineEquation>.

On the generalized Ramanujan–Nagell equation $D_1x^2-D_2=4p^n$

期刊论文
发表日期：

2026年1月27日

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Yasutsugu Fujita,
Maohua Le

摘要

Let $D_1$ $D_{1}$ and $D_2$ $D_{2}$ be positive integers and let p be an odd prime number such that $D:=D_1D_2$ $D : = D_{1} D_{2}$ is nonsquare and $\gcd (D,2p)=1$ $gcd (D, 2 p) = 1$ . Denote by $N(D_1,D_2,4,p)$ $N (D_{1}, D_{2}, 4, p)$ the number of positive integer solutions (x, n) to the generalized Ramanujan–Nagell equation $D_1x^2-D_2=4p^n$ $D_{1} x^{2} - D_{2} = 4 p^{n}$ . In this paper, we prove $N(D_1,D_2,4,p)\le 4$ $N (D_{1}, D_{2}, 4, p) \leq 4$ . Further, we prove that if the squarefree part of $D_1$ $D_{1}$ is at most 1001, then $N(D_1,D_2,4,p)\le 3$ $N (D_{1}, D_{2}, 4, p) \leq 3$ .

On the generalized Ramanujan–Nagell equation \(D_1x^2-D_2=4p^n\)

摘要