论文标题
有效解析thue-mahler方程
Efficient resolution of Thue-Mahler equations
论文作者
论文摘要
thue-mahler方程是表格$ f（x，y）= a \ cdot p_1^{z_1} \ cdots p_v^{z_v}，\ qquad \ gcd（x，x，y）= 1 $ f $是不可避免的同质$ 3 $ 3 $ 3 $ 3非零整数和$ p_1，\ dots，p_v $是理性的。解决此类方程式的现有算法需要在$ f（x，1）= 0 $的三个根中获得的数字字段中的计算。我们提供了一种新的算法，该算法仅在通过毗邻一个根获得的数字字段中进行计算，从而更适合于更高程度的示例。我们还引入了一种格子筛分技术，让人联想到Mordell-Weil筛子，这使得解决Thue-Mahler方程变得更高，并且具有更大的素数。我们提供了几个示例，包括学位$ 11美元之一。 令$ p（m）$表示整数$ m \ ge 2 $的最大素数。作为我们算法的应用，我们确定所有对$（x，y）$ coprime非负整数的应用，以便$ p（x^4-2y^4）\ le 100 $，发现精确有$ 49 $的$。
A Thue-Mahler equation is a Diophantine equation of the form $$F(X,Y) = a\cdot p_1^{z_1}\cdots p_v^{z_v}, \qquad \gcd(X,Y)=1$$ where $F$ be an irreducible homogeneous binary form of degree at least $3$ with integer coefficients, $a$ is a non-zero integer and $p_1, \dots, p_v$ are rational primes. Existing algorithms for resolving such equations require computations in the number field obtained by adjoining three roots of $F(X,1)=0$. We give a new algorithm that requires computations only in the number field obtained by adjoining one root, making it far more suited for higher degree examples. We also introduce a lattice sieving technique reminiscent of the Mordell--Weil sieve that makes it practical to tackle Thue--Mahler equations of higher degree and with larger sets of primes. We give several examples including one of degree $11$. Let $P(m)$ denote the largest prime divisor of an integer $m \ge 2$. As an application of our algorithm we determine all pairs $(X,Y)$ of coprime non-negative integers such that $P(X^4-2Y^4) \le 100$, finding that there are precisely $49$ such pairs.
