论文标题
在有限场上的立方pelt方程
On the cubic Pell equation over finite fields
论文作者
论文摘要
考虑到$ z [\ sqrt [3] {r}] $中的norm One的元素，可以将经典的pell方程扩展到立方情况，该元素满足$ x^3 + r y^3 + r y^3 + r^2 z^3-3 r x y z = 1 $。立方PEL方程的解比经典案例要难，实际上是一种解决该方程的方法，因为二磷酸方程仍缺失。在本文中，我们研究了有限磁场上的立方PEL方程，从而扩展了经典磁场的结果。特别是，在所有可能的情况下，我们提供了一种计算解决方案数量的新方法，具体取决于r的值。此外，我们还能够提供一种生成所有解决方案的方法。
The classical Pell equation can be extended to the cubic case considering the elements of norm one in $Z[\sqrt[3]{r}]$, which satisfy $x^3 + r y^3 + r^2 z^3 - 3 r x y z = 1$. The solution of the cubic Pell equation is harder than the classical case, indeed a method for solving it as Diophantine equation is still missing. In this paper, we study the cubic Pell equation over finite fields, extending the results that hold for the classical one. In particular, we provide a novel method for counting the number of solutions in all possible cases depending on the value of r. Moreover, we are also able to provide a method for generating all the solutions.
