## 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

### Keywords:

exponential Diophantine equation
Full Text:

### Online Encyclopedia of Integer Sequences:

Perfect n-th powers y^n (n >= 3) of the form x^2 + 2^a * 5^b (x, a, b >= 1, gcd(x, y) = 1).

### References:

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