Efficient resolution of Thue–Mahler equations

A Thue–Mahler equation is a Diophantine equation of the form

$$F(X,Y)=a\cdot p_1^{z_1}\cdots p_v^{z_v},\gcd <!--FUNCTION\; APPLICATION-->(X,Y)=1$$

where $F$ is an irreducible binary form of degree at least $3$ with integer coefficients, $a$ is a nonzero integer and $p_1,\dots <!--FUNCTION\; APPLICATION-->,p_v$ are rational primes. Existing algorithms for resolving such equations require computations in the field $L=\mathbb{Q}(𝜃,{𝜃}^{\prime },{𝜃}^{\prime \prime })$, where $𝜃$, ${𝜃}^{\prime }$, ${𝜃}^{\prime \prime }$ are distinct roots of $F(X,1)=0$. We give a new algorithm that requires computations only in $K=\mathbb{Q}(𝜃)$ 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 than was previously possible. 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 nonnegative integers such that $P(X^4-2Y^4)\le 100$, finding that there are precisely $49$ such pairs.