  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.
