The Diophantine equation \(a^x\pm a^y=z^n\) when \(a\) is any nonnegative integer
Volume 32, Issue 3, pp 213--221
http://dx.doi.org/10.22436/jmcs.032.03.02
Publication Date: September 13, 2023
Submission Date: October 23, 2022
Revision Date: August 01, 2023
Accteptance Date: August 05, 2023
Authors
K. Laipaporn
- School of Science, Walailak University, Nakhon Si Thammarat, 80160, Thailand.
S. Wananiyakul
- Department of Mathematics and Computer Science, Chulalongkorn University, Bangkok, 10330, Thailand.
P. Khachorncharoenkul
- School of Science, Walailak University, Nakhon Si Thammarat, 80160, Thailand.
Abstract
In this paper, all solutions of the Diophantine equation \(a^x\pm a^y=z^n\) are investigated when \(a\) is any nonnegative integer and \(n\ge 2\). In particular, if \(p\) is prime and the solutions of \(p^x+p^y=z^n\) exist, then \(p\) is either \(2\) or \(2^n-1\).
All proofs in this paper require only elementary number theory.
Share and Cite
ISRP Style
K. Laipaporn, S. Wananiyakul, P. Khachorncharoenkul, The Diophantine equation \(a^x\pm a^y=z^n\) when \(a\) is any nonnegative integer, Journal of Mathematics and Computer Science, 32 (2024), no. 3, 213--221
AMA Style
Laipaporn K., Wananiyakul S., Khachorncharoenkul P., The Diophantine equation \(a^x\pm a^y=z^n\) when \(a\) is any nonnegative integer. J Math Comput SCI-JM. (2024); 32(3):213--221
Chicago/Turabian Style
Laipaporn, K., Wananiyakul, S., Khachorncharoenkul, P.. "The Diophantine equation \(a^x\pm a^y=z^n\) when \(a\) is any nonnegative integer." Journal of Mathematics and Computer Science, 32, no. 3 (2024): 213--221
Keywords
- Diophantine equation
- Catalan's conjecture
- the fundamental theorem of arithmetic
- Mersenne prime
MSC
References
-
[1]
D. Acu, On a Diophantine equation, Gen. Math., 15 (2007), 145–148
-
[2]
M. Alan, U. Zengin, On the Diophantine equation x2 + 3a41b = yn, Period. Math. Hung., 81 (2020), 284–291
-
[3]
S. A. Arif, A. S. Al-Ali, On the Diophantine equation ax2 + bm = 4yn, Acta Arith., 103 (2002), 343–346
-
[4]
M. Buosi, A. Lemos, A. L. P. Porto, D. F. G. Santiago, On the exponential Diophantine equation px - 2y = z2 with p = k2 + 2, a prime number, Southeast-Asian J. of Sciences, 8 (2020), 103–109
-
[5]
N. Burshtein, All the solutions of the Diophantine equations px + py = z2 and px - py = z2 when p > 2 is prime, Ann. Pure Appl. Math., 11 (2019), 111–119
-
[6]
N. Burshtein, All the solutions of the Diophantine equations px + py = z4 when p > 2 is prime and x, y, z are positive integers, Ann. Pure Appl. Math., 21 (2020), 125–128
-
[7]
E. Catalan, Note extraite d’une lettre adress´ee `a l’´editeur par Mr. E. Catalan, R´ep´etiteur `a l’´ecole polytechnique de Paris, J. Reine Angew. Math., 27 (1844), 192–192
-
[8]
K. Chakraborty, A. Hoque, K. Srinivas, On the Diophantine equation cx2 + p2m = 4yn, Results Math., 76 (2021), 12 pages
-
[9]
I. Cheenchan, S. Phona, J. Ponggan, S. Tanakan, S. Boonthiem, On the Diophantine equation px + 5y = z2, SNRU J. Sci. Technol., 8 (2016), 146–148
-
[10]
M. Demırcı, On the Diophantine equation x2 + 5a pb = yn, Filomat, 31 (2017), 5263–5269
-
[11]
S. Fei, J. Luo, A note on the exponential diophantine equation (rlm2 - 1)x + (r(r - l)m2 + 1)y = (rm)z, Bull. Braz. Math. Soc. (N.S.), 53 (2022), 1499–1517
-
[12]
K. Laipaporn, S. Wananiyakul, P. Khachorncharoenkul, On the Diophantine equation 3x + p5y = z2, Walailak J. Sci. & Tech., 16 (2019), 647–653
-
[13]
P. Mihˇailescu, Primary cyclotomic units and a proof of Catalan’s conjecture, J. Reine Angew. Math., 572 (2004), 167–195
-
[14]
R. Scott, On the equations px - by = c and ax + by = cz, J. Number Theory, 44 (1993), 153–165
-
[15]
R. Scott, R. Styer, On px -qy = c and related three term exponential Diophantine equations with primes bases, J. Number Theory, 105 (2004), 212–234
-
[16]
S. Subburam, On the Ddiophantine equation lax + mby = ncz, Res. Number Theory, 4 (2018), 3 pages
-
[17]
A. Suvarnamani, Solutions of the Diophantine equation 2x + py = z2, Int. J. Math. Sci. Appl., 1 (2011), 1415–1419
-
[18]
M. Tatong, A. Suvarnamani, On the Diophantine Equation px + py = z2, In: The 15th International Conference of International Academy of Physical Sciences, Rajamangala University of Technology Thanyaburi, Thailand, (2012), 17–20
-
[19]
P. Yuan, On the Diophantine equation ax2 + by2 = ckn, Indag. Math. (N.S.), 16 (2005), 301–320
-
[20]
, GIMPS Discovers, Largest Known Prime Number, , accessed on 28 December 2021 (),