## 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.

### MSC:

 11D61 Exponential Diophantine equations 11Y50 Computer solution of Diophantine equations

### Keywords:

exponential Diophantine equation

Magma

### References:

 [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 [4] Arif, S. A., Abu Muriefah, F. S.: On a Diophantine equation. Bull. Austral. Math. Soc., 57, 1998, 189-198, · Zbl 0905.11018 [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 [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 [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 [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 [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 [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 [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 [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 [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 [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 [22] Le, M., Zhu, H.: On some generalized Lebesgue-Nagell equations. Journal of Number Theory, 131/3, 2011, 458-469, · Zbl 1219.11059 [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 [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 [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 [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
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.