INTERNATIONAL JOURNAL OF LATEST TECHNOLOGY IN ENGINEERING,
MANAGEMENT & APPLIED SCIENCE (IJLTEMAS)
ISSN 2278-2540 | DOI: 10.51583/IJLTEMAS | Volume XIII, Issue VI, June 2024
www.ijltemas.in Page 90
On the Exponential Diophantine Equation
Wariam Chuayjan, Theeradach Kaewong, and
*
Sutthiwat Thongnak
Department of Mathematics and Statistics, Faculty of Science and Digital Innovation, Thaksin University, Phatthalung
93210, Thailand.
*
Corresponding Author
DOI : https://doi.org/10.51583/IJLTEMAS.2024.130612
Received: 19 May 2024; Revised: 12 June 2024; Accepted: 17 June 2024; Published: 16 July 2024
Abstract: In this work, we determine all non-negative integers solution
to the exponential Diophantine equation
. The mathematical process bases on several theorems in Number theory, including the modular arithmetic, Divisibility
and the Division Algorithm. The solution set to the equation is
.
Keywords: divisibility; the exponential Diophantine equation; modular arithmetic; Divisibility; the Division Algorithm:
Mathematics Subject Classification: 11D61, 11D72, 11D45.
I. Introduction
The Diophantine equation is a classical problem in Number theory. For almost a decade, the famous general equation is
where and are positive integers with , and are unknown non-negative integers. In 2018, the two equations
and
, were studied (see [2]). In 2019, the equation
was proposed. They showed that the
equation has three solutions,
(see [5]). In 2020, M. Buosi et al. studied
where
prime
(see [1]). Many equations were studied from 2021 to 2022 (see [4, 6-8]). Recently, S. Tadee studied the two
equation, including
and
. He proved that
is the set of solutions to the
, and
has only a trivial solution,
(see [3]).
Previous works have shown that there is no general method for obtaining all solutions to the exponential Diophantine equations.
They must prove their work based on mathematical processes and knowledge. This challenges mathematical researchers to study
the individual equations. In this paper, we determined and proved all solutions the the exponential Diophantine equation
where , and are non-negative integers. The proof based on principles of mathematic and all solutions were given.
II. Preliminaries
In this section, we introduce basic knowledge applied in this proof.
Lemma 2.1 If is an integer, then
.
Proof: Let be an integer. We separate into even and odd.
Case 1: is even. We have where is an integer. Then, it follows that
yielding
.
Case 2: is odd. We have where is an integer. Then, it follows that
yielding
.
From two cases, we conclude that
.
Lemma 2.2 If is a positive integer, then
.
Proof: Let be a positive integer. By the Division Algorithm, there exist non-negative integers and such that
with . We obtain
.
Thus, we have
Lemma 2.3 If is an integer, then
.
Proof: Let be an integer. By the Division Algorithm, there exist integers and such that with . We
obtain
implying that
. We can write
INTERNATIONAL JOURNAL OF LATEST TECHNOLOGY IN ENGINEERING,
MANAGEMENT & APPLIED SCIENCE (IJLTEMAS)
ISSN 2278-2540 | DOI: 10.51583/IJLTEMAS | Volume XIII, Issue VI, June 2024
www.ijltemas.in Page 91
.
Thus, we have
.
III. Result
Theorem 3.1 The exponential Diophantine equation
where and are non-negative integers has two solutions,
.
Proof: Let , and be non-negative integers such that
(1)
we separate into four cases as follows.
Case 1: . By (1), we obtain . The one solution to the equation is
.
Case 2: and . From (1), we have
, impossible.
Case 3: and . From (1), we have
. (2)
From (2), if , then we have
. Thus
is a solution to the equation. If , then we have
.
This is impossible because
.
Case 4: and . Since
, (1) implies that
, thus . From (1), we can write as
.
Since
, thus we have
or
. This is impossible because of
. Therefore, the proof is complete.
IV. Conclusion
In this work, we have solved the exponential Diophantine equation
where , and are non-negative integers.
We derived three Lemmas for the proof and applied the modular arithmetic, the Divisibility, and the Division Algorithm. Finally,
we have shown that
are the solutions to the equation.
Acknowledgment
We would like to thank the reviewers for their careful reading of our manuscript and their useful comments.
References
1. Buosi, M., Lemos, A., Porto, A. L. P. and Santiago, D. F. G., (2020) On the exponential Diophantine equation
with
, a prime number, Southeast-Asian Journal of Sciences, 8(2) 103-109.
2. Rabago, J. F. T., (2018) On the Diophantine equation
where is a Prime, Thai Journal of
Mathematics, 16(3) 643-650.
3. Tadee, S., (2023) A Short Note On two Diophantine equations
and
, Journal of
Mathematics and Informatics, 24, 23-25.
4. Tadee, S., (2022) On the Diophantine equation
where is a Prime Number, Journal of
Mathematics and Informatics, 23 51-54.
5. Thongnak, S., Chuayjan, W. and Kaewong, T., (2019) On the exponential Diophantine equation
,
Southeast Asian Journal of Science, 7 (1) 1-4.
6. Thongnak, S., Chuayjan, W. and Kaewong, T., (2021) The Solution of the Exponential Diophantine Equation
, Mathematical Journal, 66 (7) 62-67.
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where and are
Non-Negative Integers, Annals of Pure and Applied Mathematics, 25 (2) 63-66.
8. Thongnak, S. Chuayjan, W. and Kaewong, T., (2022) On the Exponential Diophantine equation
,
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