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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
Software:
Magma
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References:
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