论文标题
在Lebesgue-ljunggren-Nagell类型方程式上
On a class of Lebesgue-Ljunggren-Nagell type equations
论文作者
论文摘要
给定奇怪的，coprime integers $ a $，$ b $（$ a> 0 $），我们考虑了diophantine方程$ ax^2+b^{2l} = 4y^n $，$ x，y \ in \ bbb z $，$ l \ in \ bb in \ bb n $，$ n $ odd prime，$ odd prime，$ odd prime，$ \ gcd（x，x，x，y）= 1 $ 1 $ 1 $ 1 $ 1 $ 1 $ 1 $ 1 $ 1 $ 1 $ 1 $ 1 $ 1 $ 1 $ 1 $ 1 $ 1 $ 1 $ 1 $ 1 $ 1 $ 1 $ 1 $ 1 $）在$ 2^{n-1} b^l \ equiv \ equiv \ equiv \ equiv \ equiv \ equiv \ equiv \ equiv \ pm 1（\ mod a）$和$ \ gcd（n，b）= 1 $ gcd（n，b） For other square-free integers $a>3$ and $b$ a power of an odd prime, we prove that the above Diophantine equation has no solutions for all integers $x$, $y$ with ($\gcd(x,y)=1$), $l\in\mathbb{N}$ and all odd primes $n>3$, satisfying $ 2^{n-1} b^l \ equiv \ equ \ pm 1（\ mod a）$，$ \ gcd（n，b）= 1 $，和$ \ gcd（n，h（-a））= 1 $，其中$ h（-a）$表示想象中的二次二次fiff field $ \ nath $ \ nath $ \ nathbb q（\ sq）。
Given odd, coprime integers $a$, $b$ ($a>0$), we consider the Diophantine equation $ax^2+b^{2l}=4y^n$, $x, y\in\Bbb Z$, $l \in \Bbb N$, $n$ odd prime, $\gcd(x,y)=1$. We completely solve the above Diophantine equation for $a\in\{7,11,19,43,67,163\}$, and $b$ a power of an odd prime, under the conditions $2^{n-1}b^l\not\equiv \pm 1(\mod a)$ and $\gcd(n,b)=1$. For other square-free integers $a>3$ and $b$ a power of an odd prime, we prove that the above Diophantine equation has no solutions for all integers $x$, $y$ with ($\gcd(x,y)=1$), $l\in\mathbb{N}$ and all odd primes $n>3$, satisfying $2^{n-1}b^l\not\equiv \pm 1(\mod a)$, $\gcd(n,b)=1$, and $\gcd(n,h(-a))=1$, where $h(-a)$ denotes the class number of the imaginary quadratic field $\mathbb Q(\sqrt{-a})$.
