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On the Diophantine equation $$x^2+5^a\cdot 11^b=y^n$$. (English) Zbl 1237.11019
The authors find all solutions of the title equation in integers $$a\geq 0,~b\geq 0,~n\geq 3$$, $$x>0$$ and $$y>0$$ with $$x$$ and $$y$$ coprime except when $$xab$$ is odd. When $$n=4$$, the proof is elementary, while for $$n\geq 5$$ with $$n\neq 6$$, the proof uses the theory of primitive divisors of Lucas–Lehmer sequences. The hardest case is $$n=3$$, which is reduced to finding all $$\{5,11\}$$-integral points on a collection of $$36$$ elliptic curves. Magma dealt successfully with $$35$$ of them. For the last one, the original equation is converted into a Thue–Mahler equation, which in turn is solved using linear forms in $$p$$-adic logarithms to get some huge bounds on the potential solutions, followed by LLL to reduce such bounds and a sieve to deal with the small solutions. When $$xab$$ is odd, the original equation implies that the $$n$$th term of a certain (parametric) binary recurrence is free of primes larger than $$11$$, but unfortunately the binary recurrences involved are not Lucas–Lehmer sequences, so one cannot apply the theory of primitive divisors to deduce that $$n$$ must be very small.

##### MSC:
 11D61 Exponential Diophantine equations 11Y50 Computer solution of Diophantine equations 11J86 Linear forms in logarithms; Baker’s method 11D25 Cubic and quartic Diophantine equations 11D59 Thue-Mahler equations
Magma
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