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On the diophantine equation $x^{2} + 5^{k} 17^{l} = y^{n}$

István Pink; Zsolt Rábai — 2011

Communications in Mathematics

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

Finiteness results for Diophantine triples with repdigit values

Attila Bérczes; Florian Luca; István Pink; Volker Ziegler — 2016

Acta Arithmetica

Let g ≥ 2 be an integer and $_{g} \subset ℕ$ 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 $_{g}$ .

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On the diophantine equation x 2 + 5 k 17 l = y n

Finiteness results for Diophantine triples with repdigit values

On the diophantine equation $x^{2} + 5^{k} 17^{l} = y^{n}$