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On the Diophantine equation \(x^2+2^a\cdot 5^b=y^n\). (English) Zbl 1231.11041
The authors find all the solutions of the Diophantine equation (*) \(x^2+2^a\cdot 5^b=y^n\) in positive integers \(x, y, a, b, n\) with \(x\) and \(y\) coprime and \(n\geq 3\).
Theorem. Equation (*) has no solution except for
\(n=3\qquad(x,y,a,b)\in\{(9,11,1,4), (23,9,3,2), (261,41,5,2), (383,129,7,6), (17771,681,9,2)\};\)
\(n=4\qquad(x,y,a,b)\in\{(1,3,4,1), (79,9,6,1)\};\)
\(n=5\qquad(x,y,a,b)\in\{(401,11,1,3)\};\)
\(n=6\qquad(x,y,a,b)\in\{(23,3,3,2)\};\)
\(n=8\qquad(x,y,a,b)\in\{(79,3,6,1)\}\).

MSC:
11D61 Exponential Diophantine equations
11Y50 Computer solution of Diophantine equations
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References:
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