On Two Classes of Exponential Diophantine Equations
DOI:
https://doi.org/10.26713/cma.v13i1.1676Keywords:
Catalan's conjecture, Exponential Diophantine equations, Integer solutionsAbstract
In this paper, we study on the exponential Diophantine equations: \(n^{x}+24^{y}=z^{2}\), for \(n \equiv 5\) or 7 (mod 8). We show that \(5^{x}+24^{y}=z^{2}\) has a unique positive integral solution \((2,1,7)\). Further, we show that for \(k \in \mathbb{N}\), \((8k+5)^{x}+24^{y}=z^{2}\) has a unique solution \((0,1,5)\) in non-negative integers. We also show that for a perfect square \(8m\), the exponential Diophantine equation \((8m-1)^{x}+24^{y}=z^{2}\), \(m \in \mathbb{N}\) has exactly two non-negative integral solutions \((0,1,5)\) and \((1,0,\sqrt{8m})\). Otherwise, it has a unique solution \((0,1,5)\). Finally, we illustrate our results with some examples and non-examples.
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References
D. Acu, On a Diophantine equation, General Mathematics 15 (2007), 145 – 148, URL: https://www.emis.de/journals/GM/vol15nr4/acu/acu.pdf.
S. Aggarwal, On the exponential Diophantine equation (22m+1 -1)+(13)n = z2, Engineering and Applied Science Letter 4(1) (2021), 77 – 79, DOI: 10.30538/psrp-eas12021.0064.
S.A. Arif and F.S.A. Muriefah, On the Diophantine equation x 2 + q2k+1 = yn, Journal of Number Theory 95 (2002), 95 – 100, DOI: 10.1006/jnth.2001.2750.
F. Beukers, The Diophantine equation Axp +B yq +Czr, Duke Mathematical Journal 91(1) (1998), 61 – 88, DOI: 10.1215/S0012-7094-98-09105-0.
J.J. Bravo and F. Luca, On the Diophantine equation Fn + Fm = 2 a, Quaestiones Mathematiae 39(3) (2016), 391 – 400, DOI: 10.2989/16073606.2015.1070377.
N. Burshtein, On the Diophantine equation 22x+1+7y = z2, Annals of Pure and Applied Mathematics 16(1) (2018), 177 – 179, DOI: 10.22457/apam.v16n1a19.
D.M. Burton, Elementary Number Theory, 6th edition, McGraw-Hill Education (2007), URL: https://books.google.co.in/books?hl=en&lr=&id=XMQjuoTqqRMC&oi=fnd&pg=PR9&dq=D.+M.+Burton.
Z. Cao, A note on the Diophantine equation a x + ny = cz, Acta Arthmetica XCI(1) (1999), 85 – 93, URL: http://matwbn.icm.edu.pl/ksiazki/aa/aa91/aa9115.pdf.
J. Cassels, A Diophantine equation, Glasgow Mathematical Journal 27 (1985), 11 – 18, DOI: 10.1017/S0017089500006030.
E. Catalan, Note extraite d’une lettre adressée à l’éditeur par Mr. E. Catalan, Répétiteur à l’école polytechnique de Paris, Journal für die reine und angewandte Mathematik 1844(27) (1844), p. 192, DOI: 10.1515/crll.1844.27.192.
K. Chakraborty, A. Hoque and K. Srinivas, On the Diophantine equation cx2 + p2m = 4yn, Results in Mathematics 76 (2021), Article number: 57, 10.1007/s00025-021-01366-w.
J.H.E. Cohn, On the Diophantine system x 2 -6y2 = -5, x = 2z 2 -1, Mathematica Scandinavica 82(2) (1998), 161 – 164, DOI: 10.7146/math.scand.a-13830.
J.H.E. Cohn, The Diophantine equation x2 +C = yn, Acta Arithmetica LXV(4) (1993), 387 – 381, URL: http://matwbn.icm.edu.pl/ksiazki/aa/aa65/aa6546.pdf.
W.S. Gayo and J.B. Bacan, On the Diophantine equation Mxp +(Mq +1)y = z2, European Journal of Pure and Applied Mathematics 14(2) (2021), 396 – 403, DOI: 10.29020/nybg.ejpam.v14i2.3948.
L. Hajdu and L. Szalay, On the Diophantine equations (2n -1)(6n -1) = x 2 and (an -1)(akn -1) = x2, Periodica Mathematica Hungarica 40 (2000), 141 – 145, DOI: 10.1023/A:1010335509489.
K. Ishii, On the exponential Diophantine equation (a x -1)(bn -1) = x2, Publicationes Mathematicae Debrecen 89(1-2) (2016), 253 – 256, URL: https://publi.math.unideb.hu/load_jpg.php?p=2098.
R. Keskin, A note on exponential Diophantine equation (a n - 1)(bn - 1) = x2, Proceedings - Mathematical Sciences 129 (2019), Article number 69, DOI: 10.1007/s12044-019-0520-x.
H. Kishan, M. Ravi and S. Sarita, On the Diophantine equation, Journal of Advances in Mathematics 11(9) (2016), DOI: 10.24297/jam.v11i9.815.
J. Klaska, Real-world applications of Number theory, South Bohemia Mathematical Letters 25(1) (2017), 39 – 47, URL: http://home.pf.jcu.cz/~sbml/wp-content/uploads/Klaska.pdf.
S. Kumar, S. Gupta and H. Kishan, On the Non-linear Diophantine equation 61x + 67y = z2 and 67x + 73y = z2, Annals of Pure and Applied Mathematics 18(1) (2018), 91 – 94, DOI: 10.22457/apam.v18n1a13.
M. Le (Zhanjiang), A note on the Diophantine equation x2 + by = cz, Acta Arithmetica LXXI(3) (1995), 253 – 257, URL: http://matwbn.icm.edu.pl/ksiazki/aa/aa71/aa7134.pdf.
F. Luca, On a Diophantine equation, Bulletin of the Australian Mathematical Society 61 (2000), 241 – 246, DOI: 10.1017/S0004972700022231.
P. Mihailescu, Primary cycolotomic units and a proof of Catalan’s conjecture, Journal für die reine und angewandte Mathematik 27 (2004), 167 – 195, DOI: 10.1515/crll.2004.048.
J.F.T. Rabago, On two Diophantine equations 3x +19y = z2 and 3x +91y = z2, International Journal of Mathematics and Scientific Computing 3(1) (2013), 28 – 29, URL: https://www.veltech.edu.in/wpcontent/uploads/2016/04/Paper-08-13.pdf.
M. Somanath, K. Raja, J. Kannan and A. Akila, Integral solutions of an infinite elliptic cone x2 = 9y2 +11z2, Advances and Applications in Mathematical Sciences 19(11) (2020), 1119 – 1124.
M. Somanath, K. Raja, J. Kannan and S. Nivetha, Exponential Diophantine equation in three unknowns, Advances and Applications in Mathematical Sciences 19(11) (2020), 1113 – 1118. [27] B. Sroysang, On the Diophantine equation 7x +8y = z2, International Journal of Pure and Applied Mathematics 84(1) (2013), 111 – 114, DOI: 10.12732/ijpam.v84i1.8.
B. Sury, On the Diophantine equation x2 + 2 = yn, Archiv der Mathemctik 74 (2000), 350 – 355, DOI: 10.1007/s000130050454.
A. Suvarnamani, Solution of the Diophantine equation px + qy = z2, International Journal of Pure and Applied Mathematics 94(4) (2014), 457 – 460, DOI: 10.12732/ijpam.v94i4.1.
A. Suvarnamani, A. Singta and S. Chotchaisthit, On two Diophantine equations 4x + yy = z 2 and 4 x +11y = z2, Science and Technology RMUTT Journal 1(1) (2011), 25 – 28.
N. Terai, The Diophantine equation a x + by = cz, Proceedings of the Japan Academy, Series A, Mathematical Sciences 70(1) (1994), 22 – 26, DOI: 10.3792/pjaa.70.22.
P.G. Walsh, On Diophantine equation of the form, Tatra Mountains Mathematical Publications 20(1) (2000), 87 – 89.
P. Yuan and Y. Hu, On the Diophantine equation x2 +Dm = pn, Journal of Number Theory 111(1) (2005), 144 – 153, DOI: 10.1016/j.jnt.2004.11.005.
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