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Solutions of some generalized Ramanujan-Nagell equations. (English) Zbl 1110.11012
The class of Diophantine equations considered is given by \[ x^2+D = y^n , \tag{1} \] in positive integers \(x\), \(y\), \(D\) and \(n>2\), with \(\gcd(x,y)=1\). For \(D=1\), in 1850, V.A. Lebesgue proved that there is no solution (this the first special case solved for Catalan’s equation). J. H. E. Cohn studied this equation for \(2\leq D\leq 100\) and solved completely 77 cases. Two new cases were solved by Mignotte and de Weger, using computational algebraic number theory and Baker’s theory, two other cases were solved by Bennett and Skinner using modular method. Finally the 19 remaining cases were solved by Bugeaud, Mignotte and Siksek, combining the modular method with Baker’s theory and computational algebraic number theory. Here the Authors write \[ D =p_1^{\alpha_1}\cdots p_r^{\alpha_r}=D_s D_t^2, \] where \(\alpha_1\), …\(\alpha_r\) are positive and the \(p_i\)’s are different prime numbers and \(D_s\) is square-free. Previously different authors studied the case \(r=1\).
The following result is proved:
Theorem. Supppose (1) holds with \(n>2\) and suppose that \(D\) is as above and \({}\equiv 3\pmod 4\), with \(y\) odd when \(D\equiv 7\pmod 4\). Suppose that all the \(\alpha_i\)’s are odd and all the \(p_i\)’s are \({}\equiv 3\pmod 4\). Then \(n\) is odd and every prime divisor of \(n\) divides \(3h\), where \(h\) is the class-number of the imaginary quadratic field \({\mathbb Q}(\sqrt{-D_s})\). In particular, if \(h=2^u 3^v\) with \(u\) and \(v\) non negative, then if (1) has a solution then \(n\) is a power of \(3\). Here is an example taken among many others presented in the paper: the Diophantine equation \[ x^2+3 \cdot 11^a \cdot 19^b = y^n, \] with \(a\) and \(b\) odd has no solution. The methods used in the proofs belong essentially to elementary and algebraic number theory, except for the application of the theorem on primitive divisors of linear recursive sequences of Bilu-Hanrot-Voutier (whose proof uses deeply lower bounds on linear forms of logarithms).

11D61 Exponential Diophantine equations
11J86 Linear forms in logarithms; Baker’s method
Full Text: DOI
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