On the Fermat-type Equation $x^3 + y^3 = z^p$
On the Fermat-type Equation $x^3 + y^3 = z^p$
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关于费马型方程 $x^3 y^3 = z^p$
DOI：
10.4171/cmh/386
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发表时间：
2016
期刊：
影响因子：
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通讯作者：
Nuno Freitas
中科院分区：
文献类型：
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作者：
Nuno Freitas
我们证明，当$-3$不是平方模~$p$时，费马型方程$x^3 + y^3 = z^p$没有满足$abc \ne 0$和$\gcd(a,b,c)=1$的解$(a,b,c)$。这将素数指数集的狄利克雷密度提高到大约 $0.844$，而之前的方程已知没有这样的解。为了证明，我们制定了一个独立利益标准来确定在 2 处具有某种类型的潜在良好约简的两条椭圆曲线是否具有辛或反辛同构 $p$-扭转模块。
We prove that the Fermat-type equation $x^3 + y^3 = z^p$ has no solutions $(a,b,c)$ satisfying $abc \ne 0$ and $\gcd(a,b,c)=1$ when $-3$ is not a square mod~$p$. This improves to approximately $0.844$ the Dirichlet density of the set of prime exponents to which the previous equation is known to not have such solutions. For the proof we develop a criterion of independent interest to decide if two elliptic curves with certain type of potentially good reduction at 2 have symplectically or anti-symplectically isomorphic $p$-torsion modules.