| Title:
|
On the diophantine equation $x^2+5^k17^l=y^n$ (English) |
| Author:
|
Pink, István |
| Author:
|
Rábai, Zsolt |
| Language:
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English |
| Journal:
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Communications in Mathematics |
| ISSN:
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1804-1388 |
| Volume:
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19 |
| Issue:
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1 |
| Year:
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2011 |
| Pages:
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1-9 |
| Summary lang:
|
English |
| . |
| Category:
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math |
| . |
| Summary:
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Consider the equation in the title in unknown integers $(x,y,k,l,n)$ with $x \ge 1$, $y >1$, $n \ge 3$, $k \ge 0$, $l \ge 0$ and $\gcd (x,y)=1$. Under the above conditions we give all solutions of the title equation (see Theorem 1). (English) |
| Keyword:
|
exponential diophantine equations |
| Keyword:
|
primitive divisors |
| MSC:
|
11D41 |
| MSC:
|
11D61 |
| idZBL:
|
Zbl 1264.11026 |
| idMR:
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MR2855388 |
| . |
| Date available:
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2011-10-31T08:10:44Z |
| Last updated:
|
2013-10-22 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/141674 |
| . |
| Reference:
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[1] Muriefah, F.S.A.: On the Diophantine equation $x^2+5^{2k}=y^n$.Demonstratio Mathematica 319/2 2006 285–289 Zbl 1100.11013 |
| Reference:
|
[2] Arif, S.A., Muriefah, F.S.A.: On the Diophantine equation $x^2+2^k=y^n$.Internat. J. Math. Math. Sci. 20 1997 299–304 MR 1444731, 10.1155/S0161171297000409 |
| Reference:
|
[3] Arif, S.A., Muriefah, F.S.A.: The Diophantine equation $x^2+3^m=y^n$.Internat. J. Math. Math. Sci. 21 1998 619–620 MR 1620327, 10.1155/S0161171298000866 |
| Reference:
|
[4] Arif, S.A., Muriefah, F.S.A.: The Diophantine equation $x^2+q^{2k}=y^n$.Arab. J. Sci. Sect. A Sci. 26 2001 53–62 MR 1829921 |
| Reference:
|
[5] Arif, S.A., Muriefah, F.S.A.: On the Diophantine equation $x^2+2^k=y^n$ II.Arab J. Math. Sci. 7 2001 67–71 Zbl 1010.11021, MR 1940290 |
| Reference:
|
[6] Arif, S.A., Muriefah, F.S.A.: On the Diophantine equation $x^2+q^{2k+1}=y^n$.J. Number Theory 95 2002 95–100 Zbl 1037.11021, MR 1916082, 10.1006/jnth.2001.2750 |
| Reference:
|
[7] Bennett, M.A., Ellenberg, J.S., Ng, Nathan C.: The Diophantine equation $A^4+2^dB^2=C^n$.submitted |
| Reference:
|
[8] Bennett, M.A., Skinner, C.M.: Ternary diophantine equations via Galois representations and modular forms.Canad. J. Math. 56/1 2004 23–54 Zbl 1053.11025, MR 2031121, 10.4153/CJM-2004-002-2 |
| Reference:
|
[9] Bérczes, A., Brindza, B., Hajdu, L.: On power values of polynomials.Publ. Math. Debrecen 53 1998 375–381 MR 1657483 |
| Reference:
|
[10] Bérczes, A., Pink, I.: On the diophantine equation $x^2+p^{2k}=y^n$.Archiv der Mathematik 91 2008 505–517 Zbl 1175.11018, MR 2465869 |
| Reference:
|
[11] 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 2001 75–122 Zbl 0995.11010, MR 1863855 |
| Reference:
|
[12] Bugeaud, Y.: On the diophantine equation $x^2-p^m=\pm y^n$.Acta Arith. 80 1997 213–223 MR 1451409 |
| Reference:
|
[13] Bugeaud, Y., Mignotte, M., Siksek, S.: Classical and modular approaches to exponential and diophantine equations II. The Lebesque-Nagell equation.Compos. Math. 142/1 2006 31–62 MR 2196761, 10.1112/S0010437X05001739 |
| Reference:
|
[14] 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 MR 2286850 |
| Reference:
|
[15] Bugeaud, Y., Shorey, T.N.: On the number of solutions of the generalized Ramanujan-Nagell equation.J. reine angew. Math. 539 2001 55–74 Zbl 0995.11027, MR 1863854 |
| Reference:
|
[16] Cangül, I.N., Demirci, M., Soydan, G., Tzanakis, N.: On the Diophantine equation $x^2 + 5^a11^b = y^n$.Funct. Approx. Comment. Math. 43 2010 209–225 MR 2767170 |
| Reference:
|
[17] Cangül, N., Demirci, M., Luca, F., Pintér, Á., Soydan, G.: On the Diophantine equation $x^2 + 2^a11^b = y^n$.Fibonacci Quart. 48 2010 39–46 MR 2663418 |
| Reference:
|
[18] Carmichael, R.D.: On the numerical factors of the arithmetic forms $ \alpha ^{n} \pm \beta ^{n} $.Ann. Math. (2) 15 1913 30–70 MR 1502458 |
| Reference:
|
[19] Cohn, J.H.E.: The diophantine equation $x^2+2^k=y^n$. II.Int. J. Math. Math. Sci. 22 1999 459–462 MR 1717165, 10.1155/S0161171299224593 |
| Reference:
|
[20] Cohn, J.H.E.: The diophantine equation $x^2+2^k=y^n$.Arch. Math (Basel) 59 1992 341–344 MR 1179459, 10.1007/BF01197049 |
| Reference:
|
[21] Cohn, J.H.E.: The diophantine equation $x^2+C=y^n$.Acta Arith. 65 1993 367–381 MR 1259344 |
| Reference:
|
[22] Cohn, J.H.E.: The diophantine equation $x^2+C=y^n$ II..Acta Arith. 109 2003 205–206 Zbl 1050.11038, MR 1980647, 10.4064/aa109-2-8 |
| Reference:
|
[23] Ellenberg, J.S.: Galois representations to $\mathbb {Q}$-curves and the generalized Fermat Equation $A^4+B^2=C^p$.Amer. J. Math. 126 (4) 2004 763–787 MR 2075481 |
| Reference:
|
[24] Goins, E., Luca, F., Togbe, A.: On the Diophantine Equation $x^2 + 2^{\alpha }5^{\beta }13^{\gamma } = y^n$., A.J. van der Poorten, A. Stein (eds.)ANTS VIII Proceedings ANTS VIII, Lecture Notes in Computer Science 5011 2008 430–432 Zbl 1232.11130, MR 2467863 |
| Reference:
|
[25] Győry, K., Pink, I., Pintér, Á.: Power values of polynomials and binomial Thue-Mahler equations.Publ. Math. Debrecen 65 2004 341–362 Zbl 1064.11025, MR 2107952 |
| Reference:
|
[26] Le, M.: On Cohn’s conjecture concerning the diophantine equation $x^2+2^m=y^n$.Arch. Math. Basel 78/1 2002 26–35 Zbl 1006.11013, MR 1887313 |
| Reference:
|
[27] Le, M.: On the diophantine equation $x^2+p^2=y^n$.Publ. Math. Debrecen 63 2003 27–78 Zbl 1027.11025, MR 1990864 |
| Reference:
|
[28] Lebesgue, V.A.: Sur l’impossibilité en nombres entier de l’equation $x^m=y^2+1$.Nouvelle Annales des Mathématiques (1) 9 1850 178–181 |
| Reference:
|
[29] Ljunggren, W.: Über einige Arcustangensgleichungen die auf interessante unbestimmte Gleichungen führen.Ark. Mat. Astr. Fys. 29A 1943 No. 13 Zbl 0028.10904, MR 0012090 |
| Reference:
|
[30] Ljunggren, W.: On the diophantine equation $Cx^2+D=y^n$.Pacific J. Math. 14 1964 585–596 MR 0161831, 10.2140/pjm.1964.14.585 |
| Reference:
|
[31] Luca, F.: On a diophantine equation.Bull. Austral. Math. Soc. 61 2000 241–246 Zbl 0997.11027, MR 1748703, 10.1017/S0004972700022231 |
| Reference:
|
[32] Luca, F.: On the equation $x^2+2^a3^b=y^n$.Int. J. Math. Sci. 29 2002 239–244 MR 1897992, 10.1155/S0161171202004696 |
| Reference:
|
[33] Luca, F., Togbe, A.: On the Diophantine equation $x^2 + 7^{2k} = y^n$.Fibonacci Quarterly 54 2007 322–326 Zbl 1221.11091, MR 2478616 |
| Reference:
|
[34] Luca, F., Togbe, A.: On the Diophantine equation $x^2 + 2^a5^b = y^n$.Int. J. Number Theory 4 6 2008 973–979 MR 2483306, 10.1142/S1793042108001791 |
| Reference:
|
[35] Mignotte, M., Weger, B.M.M de: On the equations $x^2+74=y^5$ and $x^2+86=y^5$.Glasgow Math. J. 38/1 1996 77–85 MR 1373962 |
| Reference:
|
[36] Mollin, R.A.: Quadratics.CRC Press, New York 1996 Zbl 0888.11041, MR 1383823 |
| Reference:
|
[37] Mordell, L.J.: Diophantine Equations.Academic Press 1969 Zbl 0188.34503, MR 0249355 |
| Reference:
|
[38] Muriefah, F.S.A., Luca, F., Togbe, A.: On the diophantine equation $x^2+5^a13^b=y^n$.Glasgow Math. J. 50 2008 175–181 MR 2381741 |
| Reference:
|
[39] Nagell, T.: Sur l’impossibilité de quelques équations a deux indeterminées.Norsk. Mat. Forensings Skifter 13 1923 65–82 |
| Reference:
|
[40] Nagell, T.: Contributions to the theory of a category of diophantine equations of the second degree with two unknowns.Nova Acta Reg. Soc. Upsal. IV Ser. 16, Uppsala 1955 1–38 MR 0070645 |
| Reference:
|
[41] Pink, I.: On the diophantine equation $x^2+2^{\alpha }3^{\beta }5^{\gamma }7^{\delta }=y^n$.Publ. Math. Debrecen 70/1–2 2007 149–166 Zbl 1121.11028, MR 2288472 |
| Reference:
|
[42] Ribenboim, P.: The little book of big primes.Springer-Verlag 1991 Zbl 0734.11001, MR 1118843 |
| Reference:
|
[43] Saradha, N., Srinivasan, A.: Solutions of some generalized Ramanujan-Nagell equations.Indag. Math. (N.S.) 17/1 2006 103–114 Zbl 1110.11012, MR 2337167, 10.1016/S0019-3577(06)80009-1 |
| Reference:
|
[44] 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, MR 2361718 |
| Reference:
|
[45] Schinzel, A., Tijdeman, R.: On the equation $y^m=P(x)$.Acta Arith. 31 1976 199–204 MR 0422150 |
| Reference:
|
[46] Shorey, T.N., van der Poorten, A.J., Tijdemann, R., Schinzel, A.: Applications of the Gel’fond-Baker method to Diophantine equations., Transcendence Theory: Advances and Applications Academic Press, London-New York, San Francisco 1977 59–77 |
| Reference:
|
[47] Shorey, T.N., Tijdeman, R.: Exponential Diophantine equations.Cambridge Tracts in Mathematics, 87. Cambridge University Press, Cambridge 1986 x+240 pp. Zbl 0606.10011, MR 0891406 |
| Reference:
|
[48] Tengely, Sz.: On the Diophantine equation $x^2+a^2=2y^p$.Indag. Math. (N.S.) 15 2004 291–304 MR 2071862 |
| Reference:
|
[49] Voutier, P.M.: Primitive divisors of Lucas and Lehmer sequences.Math. Comp. 64 1995 869–888 Zbl 0832.11009, MR 1284673, 10.1090/S0025-5718-1995-1284673-6 |
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