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Let $F \in \mathbb Z[x, y]$ be an irreducible binary form of degree $d \geq7$ and content one. Let $\alpha$ be a root of $F(x, 1)$ and assume that thefield extension $\mathbb Q(\alpha)/\mathbb Q$ is Galois. We prove that, forevery sufficiently large prime power $p^k$, the number of solutions to theDiophantine equation of Thue type $$ |F(x, y)| = tp^k $$ in integers $(x, y,t)$ such that $\gcd(x, y) = 1$ and $1 \leq t \leq (p^k)^\lambda$ does notexceed $24$. Here $\lambda = \lambda(d)$ is a certain positive, monotonouslyincreasing function that approaches one as $d$ tends to infinity. We also provethat, for every sufficiently large prime number $p$, the number of solutions tothe Diophantine equation of Thue-Mahler type $$ |F(x, y)| = tp^z $$ in integers$(x, y, z, t)$ such that $\gcd(x, y) = 1$, $z \geq 1$ and $1 \leq t \leq(p^z)^{\frac{10d - 61}{20d + 40}}$ does not exceed 1992. Our proofs follow fromthe combination of two principles of Diophantine approximation, namely thegeneralized non-Archimedean gap principle and the Thue-Siegel principle.

Absolute Bound On the Number of Solutions of Certain Diophantine Equations of Thue and Thue-Mahler Type