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A new conjecture concerning the Diophantine equation \(x^2+b^y=c^z\). (English) Zbl 1093.11022

Summary: In this paper, using a recent result of Yu. Bilu, G. Hanrot and P. M. Voutier [J. Reine Angew. Math. 539, 75–122 (2001; Zbl 0995.11010)] on primitive divisors, we prove that if \(a=|V_r|\), \(b= |U_r|\), \(c=m^2+1\), and \(b\equiv 3\pmod 4\) is a prime power, then the Diophantine equation \(x^2+b^y=c^2\) has only the positive integer solution \((x,y,z)=(a,2,r)\), where \(r>1\) is an odd integer, \(m\in\mathbb{N}\) with \(2|m\) and the integers \(U_r\), \(V_r\) satisfy \((m+\sqrt {-1})^r= V_r+U_r\sqrt{-1}\).

MSC:

11D61 Exponential Diophantine equations

Citations:

Zbl 0995.11010
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References:

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