We investigate the Lebesgue–Nagell equation <Equation ID="Equ67"> <EquationSource Format="TEX">$\begin{aligned} x^2-2=y^p \end{aligned}$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mtable> <mtr> <mtd columnalign="right"> <mrow> <msup> <mi>x</mi> <mn>2</mn> </msup> <mo>-</mo> <mn>2</mn> <mo>=</mo> <msup> <mi>y</mi> <mi>p</mi> </msup> </mrow> </mtd> </mtr> </mtable> </mrow> </math></EquationSource> </Equation>in integers x, y, p with <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$p\ge 3$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>p</mi> <mo>≥</mo> <mn>3</mn> </mrow> </math></EquationSource> </InlineEquation> an odd prime. A longstanding folklore conjecture asserts that the only solutions are the “trivial” ones with <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$y=-1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>y</mi> <mo>=</mo> <mo>-</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>. We confirm the conjecture unconditionally for <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$p\le 13$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>p</mi> <mo>≤</mo> <mn>13</mn> </mrow> </math></EquationSource> </InlineEquation>, and prove the conjecture holds for <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$p>911$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>p</mi> <mo>></mo> <mn>911</mn> </mrow> </math></EquationSource> </InlineEquation> through a careful application of lower bounds for linear forms in two logarithms. We also show that any “nontrivial” solution must satisfy <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$y > 10^{1000}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>y</mi> <mo>></mo> <msup> <mn>10</mn> <mn>1000</mn> </msup> </mrow> </math></EquationSource> </InlineEquation>. In addition, we establish auxiliary results that may support future progress on the problem, and we revisit some prior claims in the literature.

On the Lebesgue–Nagell equation $x^2-2 = y^p$

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发表日期：

2026年2月27日

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Ethan Katz,
Kyle Pratt

摘要

We investigate the Lebesgue–Nagell equation $\begin{aligned} x^2-2=y^p \end{aligned}$ $\begin{matrix} x^{2} - 2 = y^{p} \end{matrix}$ in integers x, y, p with $p\ge 3$ $p \geq 3$ an odd prime. A longstanding folklore conjecture asserts that the only solutions are the “trivial” ones with $y=-1$ $y = - 1$ . We confirm the conjecture unconditionally for $p\le 13$ $p \leq 13$ , and prove the conjecture holds for $p>911$ $p > 911$ through a careful application of lower bounds for linear forms in two logarithms. We also show that any “nontrivial” solution must satisfy $y > 10^{1000}$ $y > 10^{1000}$ . In addition, we establish auxiliary results that may support future progress on the problem, and we revisit some prior claims in the literature.

On the Lebesgue–Nagell equation \(x^2-2 = y^p\)

摘要