INTERNATIONAL JOURNAL OF LATEST TECHNOLOGY IN ENGINEERING,
MANAGEMENT & APPLIED SCIENCE (IJLTEMAS)
ISSN 2278-2540 | DOI: 10.51583/IJLTEMAS | Volume XIII, Issue V, May 2024
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On the Exponential Diophantine Equation
Theeradach Kaewong, Wariam Chuayjan, 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.2024.130516
Received: 18 May 2024; Accepted: 31 May 2024; Published: 15 June 2024
Abstract: Let and be non-negative integers. We solve the exponential Diophantine equation
. The result
indicates that the equation has a unique solution,
.
Keywords: divisibility; exponential Diophantine equation; modular arithmetic; Divisibility; Catalan’s conjecture; quadratic
residue; Legendre symbol properties:
Mathematics Subject Classification: 11D61, 11D72, 11D45.
I. Introduction
Let and be positive integers. The exponential Diophantine equation
, where and are unknown non-negative
integers, was solved by many researchers. The examples can be seen in [1, 5, 7, 9-14]. In 2023, S. Aggarwal et al. solved the two
exponential Diophantine equations, including
and
. The proof was based on the modular
arithmetic method and Catalan’s conjecture. Another equation
was proposed (see [2 - 3, 15]). Recently, the
exponential Diophantine equation
has been proposed (see [16]). They showed that the equation has no
solution. After that, S. Aggarwal et al. showed that the exponential Diophantine equation
has no solution (see
[4]). Then T. Kaewong et al. studied
. They proved that the equation has no solution (see [8]). In this work,
we solve the exponential Diophantine equation
where and are non-negative integers.
II. Preliminaries
In this section, we introduce basic knowledge applied in this proof.
Definition 2.1 [6] Let be an odd prime and
. If the quadratic congruence
has a solution, then is
said to be a quadratic residue of . Otherwise, is called a quadratic nonresidue of .
Definition 2.2 [6] Let be an odd prime and let
. The Legendre symbol
is defined by
Theorem 2.3 (Catalan’s conjecture [10]) Let , andbe integers. The Diophantine equation
with
has the unique solution
.
Theorem 2.4. [6] Let be an odd prime and let and be integers that are relatively prime to . Then the Legendre symbol has
the following properties:
(a) If
, then
.
(b)
.
(c)
.
(d)
.
(e)
and
.
INTERNATIONAL JOURNAL OF LATEST TECHNOLOGY IN ENGINEERING,
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III. Result
Theorem 3.1 The exponential Diophantine equation
where and are non-negative integers has a unique
solution
.
Proof: Let and be non-negative integers such that
. (1)
We separate into four cases as follows.
Case 1: . (1) becomes
, which is impossible.
Case 2: and . From (1) , we obtain
. Then it implies that
. This is a contradiction
because
.
Case 3: and . We have
. (2)
If , then (2) becomes
, impossible.
If , then (2) implies . From Catalan’s conjecture, it follows that
. Thus, the solution is
.
Case 4: and . From (1), we obtain
. Because
, we have
.
Then is an even positive integer, yielding or . If , then (1) becomes
, implying
. Then, is a quadratic residue of . It yields
but
. Thus,
,
yields , which is a contradiction. If , then we have
. We can see that
if is an
odd positive integer. Then, we have .
It is impossible because
. Thus, must be an even
positive integer. Let ,
. From (1), we have
or
. There exists
such that
and
where . It follows that
. (3)
Then (3) implies that or . In the case of , (3) becomes
, which is impossible. In the case
of , we can write (3) as
or
. (4)
If , then we have
, impossible. If , then (4) implies that . By Theorem 2.3, it follows that (4) has
no solution. From all cases,
is a unique non-negative integer solution to the equation. □
IV. Conclusion
We have solved the exponential Diophantine equation
where and are non-negative integers. The
knowledge in Number theory including Catalan’s conjecture, modular arithmetic, divisibility, quadratic residue and Legendre
symbol properties has been applied in the proof, we have found that the equation has a unique solution,
.
Acknowledgment
We would like to thank the reviewers for their careful reading of our manuscript and their useful comments.
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INTERNATIONAL JOURNAL OF LATEST TECHNOLOGY IN ENGINEERING,
MANAGEMENT & APPLIED SCIENCE (IJLTEMAS)
ISSN 2278-2540 | DOI: 10.51583/IJLTEMAS | Volume XIII, Issue V, May 2024
www.ijltemas.in Page 159
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