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 215
On the Exponential Diophantine Equation
2
15 17
xy
z-=
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.2024.130522
Received: 17 May 2024; Revised: 01 June 2024; Accepted: 06 June 2024; Published: 22 June 2024
Abstract: In this work, the exponential Diophantine equation
2
15 17
xy
z
, where
,xy
and
z
are non-negative integers, was
studied and presented with the theorems governing its expressions. The result indicated that
, , 0,0,0x y z
was a unique
solution to the equation.
Keywords: divisibility; exponential Diophantine equation; modular arithmetic; prime number
Mathematics Subject Classification: 11D61, 11D72, 11D45.
I. Introduction
Almost a decade ago, the exponential Diophantine equations of the form
2xy
a b z
, where
a
and
b
are positive integers,
have been studied by researchers. They solved the individual equations based on knowledge of number theory. In 2018, Rabago
[3] studied the exponential Diophantine equations
2
47
xy
z
and
. He also proved all solutions to the
equation
2
4
xy
pz
, where prime
1
q
pz
or
3 mod4p
. The Mihailescus Theorem (Catalan’s conjecture) was
applied in these proofs. In 2019, S. Thongnak et al. [7] studied the equation
2
23
xy
z
. The result was obtained by using
Mihailescus Theorem and modular arithmetic. In 2020, M. Buosi et al. [1] discovered non-negative integer solutions to
2
2
xy
pz
where prime
2
2pk
. In [2] A. Elshahed and H. Kamarulhaili proved all non-negative integer solutions to
the equation
2
4
x
ny
pz
are
, , ,x y z p
1
,1,2 1,2 1
nk nk
k

0,0,0, p
. The exponential Diophantine
equation
2
75
xy
z
was solved by S. thongnak et al (see in [8]). They proved that equation has only the trivial solution
, , 0,0,0x y z
. Recently, many exponential Diophantine equations have been solved (see in [4], [5], [9-11]).
According to previous works, the exponential Diophantine equation problem is a challenging because there is no general method
to determine solutions. In this work, we solve the exponential Diophantine equation
2
15 17
xy
z
, where
,xy
and
z
are
non-negative integers. We applied the greatest common divisor and order of modular in the proof all solutions to the equation.
II. Preliminaries
In this section, we introduce basic knowledge applied in this proof.
Definition 2.1 [6] Let
a
and
b
be given integers, with at least one of them different from zero. The greatest common divisor of
a
and
b
, denoted by
gcd ,ab
, is the positive integer
d
satisfying the following:
(a)
|da
and
|db
.
(b) If
|ca
and
|cb
, then
cd
.
Definition 2.2 [6] If
n
is a positive integer and
gcd , 1an
, the least positive integer
k
where
1 mod
k
an
is the
order of a modulo
n
denoted by
ord
n
a
.
Lemma 2.1 [6] Let
n
be a positive integer, and
a
be an integer such that
gcd , 1an
. If
ord
n
ak
and
,ij
are positive
integers then
mod
ij
a a n
if and only if
modi j k
,
,ij

.
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 216
Lemma 2.2 Let
x
be an integer. Then
2
0,1 mod3x
.
Proof: Let
x
be an integer. There exist integers
q
and
r
such that
11 , 0,1,2x q r r
. Thus, we have
0,1,2 mod3x
. It followed that
2
0,1,4 mod3x
or
2
0,1 mod3x
.
III. Result
Theorem 3.1. Let
,xy
and
z
be non-negative integers. The exponential Diophantine equation
2
15 17
xy
z
has a unique
solution:
, , 0,0,0x y z
.
Proof: Let
,xy
and
z
be non-negative integers such that
2
15 17
xy
z
. (1)
For convenience, let us consider four cases as follows.
Case 1:
0x
and
0y
. From (1), we have
2
0z
or
0z
, so the solution is
0,0,0
.
Case 2:
0x
and
0y
. (1) becomes
2
17
y
z
, impossible, because of
1 7 0
y

.
Case 3:
0x
and
0y
. From (1), we have
2
15 1
x
z 
, which implies that
2
2 mod3z
, which contradicts to Lemma
2.1.
Case 4:
0x
and
0y
, (1) implies that
2
1 1 mod4
x
z
. Since
2
2 mod4z 
,
x
is an even positive integer.
Let
2,x k k
. By (1), we have
22
17 15
yk
z
or
17 15 15
y k k
zz
. There exists
0,1,2,..., y
,
such that
15 17
k
z

and
15 17
ky
z

where
y


. We obtain
2 15 17 17
ky

or
2
2 3 5 17 1 17
k k y

. (2)
Because of
17 | 2 3 5
kk

and (2), we have
0
. It follows that
2 3 5 1 17
k k y
.
(3)
(3) implies that
1 1 0 mod3
y
, so
y
is an odd positive integer. By (3) again, we obtain that
2 1 mod5
y

, then
2
2 2 mod5
y
. Due to
5
ord 2 4
, we applied Lemma 2.1. Then, we obtain
2 mod4y
. There exists
l
such
that
2 4 2 1 2y l l
, which implies that
2|y
. This is a contradiction because
y
is an odd positive integer.
IV. CONCLUSION
We have solved the exponential Diophantine equation
2
15 17
xy
z
, where
,xy
and
z
are non-negative integers. This work
was done using knowledge of Number Theory, including the greatest common divisor and order of modulo. The result shows that
the equation has a unique solution (trivial solution),
, , 0,0,0x y z
.
Acknowledgment
We would like to thank the reviewers for their careful reading of our manuscript and their useful comments.
References
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2
xy
pz
with
2
2pk
, a prime number, Southeast-Asian Journal of Sciences, 8(2) 103-109.
2. Elshahed, A. and Kamarulhaili, H, (2020) On the exponential Diophantine equation
2
4
x
ny
pz
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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 217
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