Efficient resolution of Thue-Mahler equations

Abstract

A Thue-Mahler equation is a Diophantine equation of the form $$F(X,Y)=a\cdot p_1^{z_1}\cdots p_v^{z_v},\gcd (X,Y)=1$$ 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.