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The Diophantine equation \(x^2+b^y=c^z\). (English) Zbl 0987.11020

Let \(m\), \(r\) be positive integers such that \(2\mid m\), \(2\nmid r\) and \(r>1\). Let \(U_r\), \(V_r\) be integers with \(V_r+ U_r \sqrt{-1}= (m+\sqrt{-1})^r\). In this paper the authors prove that if \(a=|V_r|\), \(b=|U_r|\), \(c=m^2+1\), \(b> 8\cdot 10^6\) and \(b\) is a prime with \(b\equiv 3\pmod 4\), then the equation \(x^2+ b^y= c^z\) has only the positive integer solution \((x,y,z)= (a,2,r)\).
{Reviewer’s remark: The conjecture proposed by the authors is false}.

MSC:

11D61 Exponential Diophantine equations
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