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An exponential Diophantine equation. (English) Zbl 0981.11013
The author gives all the positive integer solutions \((x,y,m,n)\) of the equation \[ x^2+p^{2m}=y^n, \quad \text{\(x\), \(y\), \(m\), \(n\in {\mathbb N}\), \(\gcd(x,y)=1\), \(n>2\)} \] satisfying \(2 |n\) or \(2\nmid n\) and \(p\not\equiv (-1)^{(p-1)/2} \pmod{4n}\), where \(p\) is a prime number \({}>3\). The proof combines many results on several Diophantine equations. In particular, it uses a new deep result of Bilu-Hanrot-Voutier on primitive divisors of Lucas and Lehmer numbers.

MSC:
11D61 Exponential Diophantine equations
11B39 Fibonacci and Lucas numbers and polynomials and generalizations
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