Some exponential Diophantine equations. II: The equation \(x^2 - dy^2=k^z\) for even \(k\). (English) Zbl 1491.11034
Authors’ abstract: Let \(D\) be a nonsquare integer, and let \(k\) be an integer with \(|k|\geq 1\) and \(\gcd(D,k)=1\). In the part \(I\) of this paper, using some properties on the representation of integers by binary quadratic primitive forms with discriminant \(4D\), the second author [Part I, J. Number Theory 55, No. 2, 209–221 (1995; Zbl 0852.11015)] gave a series of explicit formulas for the integer solutions \((x, y, z)\) of the equation \(x^2-Dy^2=k^z\), \(\gcd(x,y)=1\), \(z>0\) for \(2\nmid k\) or \(|k|\) is a power of 2. In this part, we give similar results for the other cases of \(k\).
MSC:
| 11D61 | Exponential Diophantine equations |
Keywords:
polynomial-exponential Diophantine equation; explicit formula for integer solutions; representation of integer by binary quadratic primitive formCitations:
Zbl 0852.11015References:
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