Efficient resolution of Thue-Mahler equations
Adela Gherga, Samir Siksek
TL;DR
The paper develops a practical algorithm for solving Thue–Mahler equations F(X,Y)=a·∏p_i^{z_i} by working in the single root field K=Q(θ) rather than larger extensions, and by integrating a lattice-based reduction with a Dirichlet-type sieve. The method reduces ideal equations to a single principal form and translates the problem into S-unit equations using an explicit, well-chosen S-unit basis, avoiding p-adic field computations. It combines explicit lower bounds for linear forms in both p-adic and real/complex logarithms (Matveev–Yu) with multi-embedding approximate relations to obtain progressively tighter bounds on exponents, followed by a sieve that prunes the search space via lattice cosets. The approach enables solving higher-degree Thue–Mahler equations and large-prime sets, as demonstrated by degree-11 examples and the P(X^4−2Y^4)≤100 application, highlighting significant practical gains over prior methods.
Abstract
A Thue-Mahler equation is a Diophantine equation of the form $$F(X,Y) = a\cdot p_1^{z_1}\cdots p_v^{z_v}, \qquad \gcd(X,Y)=1$$ 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.