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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).

MSC:

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

[1] Arif, S. A.; Abu Muriefah, F. S., The Diophantine equation \(x^2 + 3^m = y^{n\) · Zbl 0905.11017
[2] Bennett, M. A.; Skinner, C. M., Ternary Diophantine equations via Galois representations and modular forms, Canad. J. Math., 56, 1, 23-54 (2004) · Zbl 1053.11025
[3] Bilu, Y., On Le’s and Bugeaud’s papers about the equation \(ax^2 + b^{2m−1} = 4 c^P \), Monatsh. Math., 137, 1-3 (2002) · Zbl 1012.11023
[4] Bilu, Y.; Hanrot, G.; Voutier, P. M., Existence of primitive divisors of Lucas and Lehmer numbers (with an appendix by M. Mignotte), J. Reine Angew. Math., 539, 75-122 (2001) · Zbl 0995.11010
[5] Bugeaud, Y., On some exponential Diophantine equations, Monatsh. Math., 132, 93-97 (2001) · Zbl 1014.11023
[6] Bugeaud Y., Mignotte M., Siksek S. — Classical and modular approaches to exponential and Diophantine equations II. The Lebesgue-Nagell equation, math.NT/0405220.; Bugeaud Y., Mignotte M., Siksek S. — Classical and modular approaches to exponential and Diophantine equations II. The Lebesgue-Nagell equation, math.NT/0405220. · Zbl 1128.11013
[7] Bugeaud, Y.; Shorey, T. N., On the number of solutions of the generalized Ramanujan-Nagell equation, I. Reine Angew. Math., 539, 55-74 (2001) · Zbl 0995.11027
[8] Cohn, J. H.E., The Diophantine equation \(x^2 + C = y^n\), Acta Arithmetica, 55, 367-381 (1993) · Zbl 0795.11016
[9] Darmon, H.; Granville, A., On the equations \(z^m = F\)(x, y) and \(Ax^p + By^q = Cz^r \), Bull. London Math. Soc., 27, 513-543 (1995) · Zbl 0838.11023
[10] Maohua, Le, On the number of solutions of the Diophantine equation \(x^2 + D = p^n\), C. R. Acad. Sci. Paris Sér. A, 317, 135-138 (1993) · Zbl 0788.11013
[11] Maohua, Le, A note on the generalised Ramanujan-Nagell equation, J. Number Theory, 50, 193-201 (1995) · Zbl 0821.11020
[12] Maohua, Le, Some exponential Diophantine equations. I. The equation \(D_1x^2\) − \(D_2y^2 = λ k^z\), J. Number Theory, 55, 209-221 (1995) · Zbl 0852.11015
[13] Maohua, Le, On the Diophantine equation \(D_1x^2 + D_2^m = 4 y^m\), Monatsh. Math., 120, 121-125 (1995) · Zbl 0877.11020
[14] Lebesgue, V. A., Sur l’impossibilité en nombres entiers de l’équation \(x^m = y^2 + 1\), Nouvelles Annales des Mathématiques (1), 9, 178-181 (1850)
[15] Luca, F., On a Diophantine equation, Bull. Austral. Math. Soc., 61, 2, 241-246 (2000) · Zbl 0997.11027
[16] Luca, F., On the equation \(x^2 + 2^a\) · \(3^b = y^{n\) · Zbl 1085.11021
[17] Mignotte, M.; de Weger, B. M.M., On the equations \(x^2 +74 = y^5\) and \(x^2 + 86 = y^5\), Glasgow Math. J., 38, 1, 77-85 (1996) · Zbl 0847.11011
[18] Mollin, R. A., Quadratics (1996), CRC Press: CRC Press New York, 387 p. · Zbl 0858.11001
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