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Applications of the hypergeometric method to the generalized Ramanujan-Nagell equation. (English) Zbl 1010.11020

A Ramanujan-Nagell equation, according to the definition that the authors adopt, is one of the shape \(f(x)=p_1^{n_1}\cdots p_r^{n_r}\), where \(f(x)\) is a polynomial with integral coefficients and at least two simple zeros, \(p_1,\ldots, p_r\) are distinct rational primes and \(n_1,\ldots,n_r\geq 2\) are integers. The integer unknowns of the equation are \(x, n_1,\ldots,n_r\). First, the authors obtain sharp rational approximations to square roots of integers. Their main results are too technical to be stated here. The corollaries to these results deal with approximations of the shape \[ |\sqrt{y}-{p\over sq}|> q^{-\lambda(y)} , \] where \(s\) is given (a special important instance is \(s=1\)) and \(q\) is assumed to be a power of \(y\); the precise results are still somewhat too long to be stated here. The applications of these results to special historical Ramanujan-Nagell equations, in combination with previous results due to various authors, are remarkable: For example, speaking about \(x^2+D=y^n\), where \(y\) has a prescribed value chosen from a certain long list of values, they obtain explicit upper bounds for \(n\) in terms of \(y\) and \(D\). Also, using one of their two main technical results, they can show that, if \(D\) is a positive integer and \(p\) is an odd prime not dividing \(D\), then the Diophantine equation \(x^2-D=p^n\) has at most three solutions in positive integers \(x,n\) and there are infinite families of pairs \((D,p)\) for which there are three solutions.
Their method, which is very technical, uses Padé approximations to \(\sqrt{1-z}\), where the approximants are obtained by means of certain contour integrals. At the end the authors discuss the possibility of applying their method to Ramanujan-Nagell equations when \(f(x)\) is not necessarily of the form \(x^2\pm D\).

MSC:

11D61 Exponential Diophantine equations
11D45 Counting solutions of Diophantine equations
11J82 Measures of irrationality and of transcendence
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