Given a binary recurrence ${u_n}_{n≥0}$, we consider the Diophantine equation $u_{n_1}^{x_1} ⋯ u_{n_L}^{x_L} = 1$ with nonnegative integer unknowns $n_1,..., n_L$, where $n_i ≠ n_j$ for 1 ≤ i < j ≤ L, $max{|x_i|: 1 ≤ i ≤ L} ≤ K$, and K is a fixed parameter. We show that the above equation has only finitely many solutions and the largest one can be explicitly bounded. We demonstrate the strength of our method by completely solving a particular Diophantine equation of the above form.
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Let g ≥ 2 be an integer and $𝓡_g ⊂ ℕ$ be the set of repdigits in base g. Let $𝓓_g$ be the set of Diophantine triples with values in $𝓡_g$; that is, $𝓓_g$ 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 $𝓡_g$. We prove effective finiteness results for the set $𝓓_g$.
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