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Consider the equation in the title in unknown integers $(x,y,k,l,n)$ with $x\ge 1$, $y>1$, $n\ge 3$, $k\ge 0$, $l\ge 0$ and $gcd(x,y)=1$. Under the above conditions we give all solutions of the title equation (see Theorem 1).

Let g ≥ 2 be an integer and ${}_g\subset \mathbb{N}$ be the set of repdigits in base g. Let ${}_g$ be the set of Diophantine triples with values in ${}_g$; that is, ${}_g$ 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 ${}_g$. We prove effective finiteness results for the set ${}_g$.