Let g ≥ 2 be an integer and be the set of repdigits in base g. Let be the set of Diophantine triples with values in ; that is, is the set of all triples (a,b,c) ∈ ℕ³ with c < b < a such that ab + 1, ac + 1 and bc + 1 lie in the set . We prove effective finiteness results for the set .
Let A be an arbitrary integral domain of characteristic 0 that is finitely generated over ℤ. We consider Thue equations F(x,y) = δ in x,y ∈ A, where F is a binary form with coefficients from A, and δ is a non-zero element from A, and hyper- and superelliptic equations in x,y ∈ A, where f ∈ A[X], δ ∈ A∖0 and . Under the necessary finiteness conditions we give effective upper bounds for the sizes of the solutions of the equations in terms of appropriate representations for A, δ, F, f, m. These...
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