پیوندهای مفید
آمار
| تعداد نشریات | 26 |
| تعداد شماره‌ها | 399 |
| تعداد مقالات | 3,523 |
| تعداد مشاهده مقاله | 5,417,149 |
| تعداد دریافت فایل اصل مقاله | 3,702,973 |
On the solution of the exponential Diophantine equation 2x+m2y=z2, for any positive integer m | ||
| Journal of Hyperstructures | ||
| دوره 11، شماره 2، اسفند 2022، صفحه 329-337 اصل مقاله (273.59 K) | ||
| نوع مقاله: Research Paper | ||
| شناسه دیجیتال (DOI): 10.22098/jhs.2023.2591 | ||
| نویسندگان | ||
| Mridul Dutta* 1؛ Padma Bhushan Borah2 | ||
| 1Department of Mathematics, Dudhnoi College, P.O. Dudhnoi, Goalpara, Assam, India | ||
| 2Department of Mathematics, Gauhati University, Guwahat, Assam, India | ||
| چکیده | ||
| It is well known that the exponential Diophantine equation 2x+ 1=z2 has the unique solution x=3 and z=3 in non-negative integers, which is closely related to the Catlan's conjecture. In this paper, we show that for m∈N, m>1, the exponential Diophantine equation 2x+m2y=z2 admits a solution in positive integers (x, y,z) if and only if m=2αMn, α≠0 for some Mersenne number Mn. When m=2αMn, α≠0, the unique solution is (x,y,z)=(2+n+2α,1, 2α(2n+1)). Finally, we conclude with certain examples and non-examples alike! The novelty of the paper is that we mainly use elementary methods to solve a particular class of exponential Diophantine equations. | ||
| کلیدواژه‌ها | ||
| Mersenne numbers؛ Catalan's Conjecture؛ Exponential Diophantine equations | ||
| مراجع | ||
|
[1] A. Suvarnamani, Solution of the Diophantine equation px +qy = z2 , Int. J. Pure.Appl. Math., 94(4) (2014), 457-460. [2] B. Sroysang, On the Diophantine equation 7x + 8y = z2 , Int. J. Pure. Appl.Math., 84 (1) (2013), 111-114. [3] D. Acu, On a Diophantine equation 2x+5y = z2 , Gen. Math., 15 (2007), 145-148. [4] D. M. Burton, Elementary number theory, McGraw-Hill Education, (2006). [5] E. Catalan, Note extraite d´une lettre adressee `a l´editeur, J. Reine Angew. Math.,27 (1844), 192. [6] F. Luca, On a Diophantine equation, Bull. Austral. Math. Soc., 61 (2000), 241-246. [7] J. F. T. Rabago, On the Diophantine equations 3x + 19y = z2 , 3 x + 91y = z2 , Int. J. of Mathematics and Scientific Computing, 3(1) (2013), 28-29. [8] J. H. E. Cohn , The Diophantine equation x2 + c = yn , Acta Arith., LXV. 4 (1993). [9] J. J. Bravo and F. Luca, On the Diophantine equation Fn + Fm = 2a , Quaest. Math., 39(3) (2016). [10] M. Dutta and P. B. Borah, On the solution of a class of exponential Diophantine equations, South East Asian J. Math. Math. Sci., 18(3) (2022). [11] M. Somanath, K. Raja, J. Kannan and A. Akila, Integral Solutions of an infinite Elliptic Cone x2 = 9y2+ 11z2 , Adv. Appl. Math. Sci., 19(11) (2020), 1119-1124. [12] N. Burshtein, On the Diophantine equation 22x+1 + 7y = z2 , Ann. Pure Appl.Math., 16 (1) (2018), 177-179. [13] P. B. Borah and M. Dutta, On two classes of exponential Diophantine equations,Communications in Mathematics and Applications, 13 (1) (2022), 137145. [14] P. B. Borah and M. Dutta, On the Diophantine equation 7x + 32y = z2 and Its generalization, Integers, 22 (2) (2022). [15] P. Mihailescu, Primary cycolotomic units and a proof of Catalans conjecture, J. Reine Angew. Math., 27(2004),167-195. [16] S. A. Arif and F. S. A. Muriefah, On the Diophantine equation x2 + q2k+1 = yn , J. Number Theory, 95 (2002), 95-100. [17] S. Kumar, S. Gupta and H. Kishan, On the Non-Linear Diophantine equation 61x + 67y = z2 , 67x + 73y = z2 , Ann. Pure Appl. Math., 18(1) (2018), 91-94. [18] W. S. Gayo and J. B. Bacan, On the Diophantine equation Mxp + (Mq + 1)y = z2 , Eur. J. Pure Appl. Math., 14 (2) (2021), 396-403. | ||
|
آمار تعداد مشاهده مقاله: 268 تعداد دریافت فایل اصل مقاله: 429 |
||