On the Diophantine equation \(x^2 + 2^{\alpha} 5^{\beta} 17^{\gamma} = y^n\). (English) Zbl 1332.11041
All integer solutions \((x,y,n,\alpha, \beta,\gamma)\) of the equation \(x^2+ 2^\alpha 5^\beta 17^\gamma= y^n\), \(\min\{x,y\}\geq 1\), \(\gcd(x,y)= 1\), \(n\geq 3\), \(\min\{\alpha,\beta,\gamma\}\geq 0\) are given.
Reviewer: Le Maohua (Zhanjiang)
Keywords:
exponential Diophantine equationSoftware:
MagmaReferences:
| [1] | Muriefah, F. S. Abu: On the Diophantine equation \(x^2+5^{2k}=y^n\). Demonstratio Math., 39, 2006, 285-289, · Zbl 1100.11013 |
| [2] | Muriefah, F. S. Abu, Arif, S. A.: The Diophantine equation \(x^2+5^{2k+1}=y^n\). Indian J. Pure Appl. Math., 30, 1999, 229-231, · Zbl 0940.11017 |
| [3] | Muriefah, F. S. Abu, Luca, F., Togbé, A.: On the Diophantine equation \(x^2+5^a\cdot 13^b=y^n\). Glasgow Math. J., 50, 2006, 175-181, · Zbl 1186.11016 · doi:10.1017/S0017089507004028 |
| [4] | Arif, S. A., Abu Muriefah, F. S.: On a Diophantine equation. Bull. Austral. Math. Soc., 57, 1998, 189-198, · Zbl 0905.11018 · doi:10.1017/S0004972700031580 |
| [5] | Arif, S. A., Abu Muriefah, F. S.: On the Diophantine equation \(x^2+2^k=y^n\). Int. J. Math. Math. Sci., 20, 1997, 299-304, · Zbl 0881.11038 · doi:10.1155/S0161171297000409 |
| [6] | Arif, S. A., Abu Muriefah, F. S.: On the Diophantine equation \(x^2+3^m=y^n\). Int. J. Math. Math. Sci., 21, 1998, 619-620, · Zbl 0905.11017 · doi:10.1155/S0161171298000866 |
| [7] | Arif, S. A., Abu Muriefah, F. S.: On the Diophantine equation \(x^2+q^{2k+1}=y^n\). J. Number Theory, 95, 2002, 95-100, · Zbl 1037.11021 · doi:10.1006/jnth.2001.2750 |
| [8] | Bérczes, A., Brindza, B., Hajdu, L.: On power values of polynomials. Publ. Math. Debrecen, 53, 1998, 375-381, · Zbl 0911.11019 |
| [9] | Bérczes, A., Pink, I.: On the diophantine equation \(x^2 + p^{2k} = y^n\). Archiv der Mathematik, 91, 2008, 505-517, · Zbl 1175.11018 · doi:10.1007/s00013-008-2847-x |
| [10] | Bérczes, A., Pink, I.: On the diophantine equation \(x^2 + p^{2l+1} = y^n\). Glasgow Math. Journal, 54, 2012, 415-428, · Zbl 1266.11059 · doi:10.1017/S0017089512000067 |
| [11] | Bilu, Yu., Hanrot, G., Voutier, P.: Existence of primitive divisors of Lucas and Lehmer numbers (with an appendix by M. Mignotte). J. Reine Angew. Math., 539, 2001, 75-122, · Zbl 0995.11010 · doi:10.1515/crll.2001.080 |
| [12] | Bugeaud, Y., Muriefah, F. S. Abu: The Diophantine equation \(x^2 + c = y^n\): a brief overview. Revista Colombiana de Matematicas, 40, 2006, 31-37, · Zbl 1189.11019 |
| [13] | Bugeaud, Y., Mignotte, M., Siksek, S.: Classical and modular approaches to exponential Diophantine equations. II. The Lebesgue-Nagell Equation. Compositio Math., 142, 2006, 31-62, · Zbl 1128.11013 · doi:10.1112/S0010437X05001739 |
| [14] | Cangül, I. N., Demirci, M., Inam, I., Luca, F., Soydan, G.: On the Diophantine equation \(x^2 + 2^a\cdot 3^b\cdot 11^c = y^n\). Accepted to appear in Math. Slovaca. · Zbl 1349.11069 |
| [15] | Cangü, I. N., Demirci, M., Luca, F., Pintér, A., Soydan, G.: On the Diophantine equation \(x^2 + 2^a\cdot 11^b = y^n\). Fibonacci Quart., 48, 2010, 39-46, · Zbl 1219.11056 |
| [16] | Cangül, I. N., Demirci, M., Soydan, G., Tzanakis, N.: The Diophantine equation \(x^2+5^a\cdot 11^b=y^n\). Funct. Approx. Comment. Math, 43, 2010, 209-225, · Zbl 1237.11019 |
| [17] | Cannon, J., Playoust, C.: MAGMA: a new computer algebra system. Euromath Bull., 2, 1, 1996, 113-144, |
| [18] | Goins, E., Luca, F., Togbé, A.: On the Diophantine equation \(x^2 + 2^{\alpha }5^{\beta }13^{\gamma } = y^n\). Algorithmic number theory, Lecture Notes in Computer Science, 5011/2008, 2008, 430-442, · Zbl 1232.11130 · doi:10.1007/978-3-540-79456-1_29 |
| [19] | Ko, C.: On the Diophantine equation \(x^2=y^n+1\), \(xy\not =0\). Sci. Sinica, 14, 1965, 457-460, · Zbl 0163.04004 |
| [20] | Le, M.: An exponential Diophantine equation. Bull. Austral. Math. Soc., 64, 2001, 99-105, · Zbl 0981.11013 · doi:10.1017/S0004972700019717 |
| [21] | Le, M.: On Cohn’s conjecture concerning the Diophantine \(x^2+2^m=y^n\). Arch. Math. (Basel), 78, 2002, 26-35, · Zbl 1006.11013 · doi:10.1007/s00013-002-8213-5 |
| [22] | Le, M., Zhu, H.: On some generalized Lebesgue-Nagell equations. Journal of Number Theory, 131/3, 2011, 458-469, · Zbl 1219.11059 · doi:10.1016/j.jnt.2010.09.009 |
| [23] | Lebesgue, V. A.: Sur l’impossibilité en nombres entiers de l’equation \(x^m=y^2+1\). Nouv. Annal. des Math., 9, 1850, 178-181, |
| [24] | Luca, F.: On a Diophantine equation. Bull. Austral. Math. Soc., 61, 2000, 241-246, · Zbl 0997.11027 · doi:10.1017/S0004972700022231 |
| [25] | Luca, F.: On the Diophantine equation \(x^2+2^a\cdot 3^b=y^n\). Int. J. Math. Math. Sci., 29, 2002, 239-244, · Zbl 1085.11021 |
| [26] | Luca, F., Togbé, A.: On the Diophantine equation \(x^2+2^a\cdot 5^b=y^n\). Int. J. Number Theory, 4, 2008, 973-979, · Zbl 1231.11041 · doi:10.1142/S1793042108001791 |
| [27] | Luca, F., Togbé, A.: On the Diophantine equation \(x^2 + 7^{2k} = y^n\). Fibonacci Quart., 54, 4, 2007, 322-326, · Zbl 1221.11091 |
| [28] | Pink, I., Rábai, Zs.: On the Diophantine equation \(x^2 + 5^{k}17^l = y^n\). Commun. Math., 19, 2011, 1-9, · Zbl 1264.11026 |
| [29] | Saradha, N., Srinivasan, A.: Solutions of some generalized Ramanujan-Nagell equations. Indag. Math. (N.S.), 17/1, 2006, 103-114, · Zbl 1110.11012 · doi:10.1016/S0019-3577(06)80009-1 |
| [30] | Saradha, N., Srinivasan, A.: Solutions of some generalized Ramanujan-Nagell equations via binary quadratic forms. Publ. Math. Debrecen, 71/3-4, 2007, 349-374, · Zbl 1164.11020 |
| [31] | Schinzel, A., Tijdeman, R.: On the equation \(y^m = P(x)\). Acta Arith., 31, 1976, 199-204, · Zbl 0339.10018 |
| [32] | Shorey, T. N., Tijdeman, R.: Exponential Diophantine equations, Cambridge Tracts in Mathematics. 1986, 87. Cambridge University Press, Cambridge, x+240 pp.. · Zbl 0606.10011 |
| [33] | Tengely, Sz.: On the Diophantine equation \(x^2+a^2=2y^p\). Indag. Math. (N.S.), 15, 2004, 291-304, · Zbl 1088.11021 · doi:10.1016/S0019-3577(04)90021-3 |
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