OnS-unit equations in two unknowns

On the number of solutions of polynomial congruences and Thue equations

“…Proof. This is Theorem 1 of [17] together with the observation that we may take the triples in A from (&'K \ {0})3 .…”

“…Recently, Evertse, Gyory, Stewart, and Tijdeman [17] proved that almost all equivalence classes of S-unit equations of the form (84) have very few solutions and their result is our next lemma. Proof.…”

“…In the next section we shall make use of such a reduction to study the Thue-Mahler equation and the generalized Ramanujan-Nagell equation. We shall also appeal to an eWective version of Lemma 7 established in [17]. …”

“…In the second part we shall address the problem of finding forms F for which (1) has many solutions for arbitrarily large integers h. Finally we shall obtain upper bounds for the number of solutions of certain Thue-Mahler and Ramanujan-Nagell equations by appealing to estimates of Evertse, Gyory, Stewart, and Tijdeman [17] for the number of solutions of S-unit equations.…”

S-unit equations and their applications

“…Theorem 5 (Evertse, Gyory, Stewart, Tijdeman [26] In this section we shall describe some ideas behind the proofs of Theorems 1 and 2. By the subspace Theorem, the solutions of (4.6) in x E pn-1 (K) are contained in finitely many proper projective subspaces of pn-l (K are finite, we obtain that the total number of solutions of ( 4.3) subject to ( 4.4) is finite.…”