An application of the symplectic argument to some Fermat-type equations
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Bibliographic record
Abstract
Let p be a prime number. In the early 2000s, it was proved that the Fermat equations with coefficients <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="block" overflow="scroll"> <mml:mrow> <mml:mn>3</mml:mn> <mml:msup> <mml:mrow> <mml:mi>x</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>p</mml:mi> </mml:mrow> </mml:msup> <mml:mo>+</mml:mo> <mml:mn>8</mml:mn> <mml:msup> <mml:mrow> <mml:mi>y</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>p</mml:mi> </mml:mrow> </mml:msup> <mml:mo>+</mml:mo> <mml:mn>21</mml:mn> <mml:msup> <mml:mrow> <mml:mi>z</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>p</mml:mi> </mml:mrow> </mml:msup> <mml:mo>=</mml:mo> <mml:mn>0</mml:mn> <mml:mspace width="1em"/> <mml:mtext> and </mml:mtext> <mml:mspace width="1em"/> <mml:mn>3</mml:mn> <mml:msup> <mml:mrow> <mml:mi>x</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>p</mml:mi> </mml:mrow> </mml:msup> <mml:mo>+</mml:mo> <mml:mn>4</mml:mn> <mml:msup> <mml:mrow> <mml:mi>y</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>p</mml:mi> </mml:mrow> </mml:msup> <mml:mo>+</mml:mo> <mml:mn>5</mml:mn> <mml:msup> <mml:mrow> <mml:mi>z</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>p</mml:mi> </mml:mrow> </mml:msup> <mml:mo>=</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> </mml:math> do not admit non-trivial solutions for a set of exponents p with Dirichlet density 1/4 and 1/8, respectively. In this note, using a recent criterion to decide if two elliptic curves over <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mi mathvariant="double-struck">Q</mml:mi> </mml:math> with certain types of additive reduction at 2 have symplectically isomorphic p -torsion modules, we improve these densities to 3/8.
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| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.001 | 0.001 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
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| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.000 | 0.000 |
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