Over the past decade, significant research has been conducted on the equation $ a^x+b^y = z^2 $ under various conditions imposed on $ a $ and $ b $ or on $ x $ and $ y $. Most studies focus on conditions where the equation has no solution, while some explore cases with infinitely many solutions, often considering scenarios where $ x $ or $ y $ is even. Motivated by this line of inquiry, we have been inspired to investigate and analyze equations of the form $ p^x+ q^{2y} = z^{2 n} $ for two distinct primes $ p $ and $ q $, and to present explicit forms of their solutions $ (p, x, q, y, z, n) $. Recent studies on the exponential Diophantine equation $ p^x+q^y = z^2 $, where $ p $ and $ q $ are primes, have addressed cases where $ p = 2 $ or $ p\equiv q\pmod 4 $. In this paper, we address the case where $ p\not\equiv q\pmod 4 $ and $ y $ is even. In addition, we explore special cases where $ z $ is the prime and provide the complete set of solutions for $ p^x+q^{2y} = z^{2n} $. We also show that the equation has no solution when $ \{2, 3\}\nsubseteq\{p, q, z\} $. In other words, we provide almost explicit solutions to $ p^x+ q^{y} = z^{2 n} $ except for the case where both $ x $ and $ y $ are odd.
Citation: Kittipong Laipaporn, Saeree Wananiyakul, Prathomjit Khachorncharoenkul. Explicit solutions and non-solutions for the Diophantine equation $ p^x+ q^{2y} = z^{2 n} $ involving primes $ p\not\equiv q \pmod 4 $[J]. AIMS Mathematics, 2025, 10(7): 15720-15736. doi: 10.3934/math.2025704
Over the past decade, significant research has been conducted on the equation $ a^x+b^y = z^2 $ under various conditions imposed on $ a $ and $ b $ or on $ x $ and $ y $. Most studies focus on conditions where the equation has no solution, while some explore cases with infinitely many solutions, often considering scenarios where $ x $ or $ y $ is even. Motivated by this line of inquiry, we have been inspired to investigate and analyze equations of the form $ p^x+ q^{2y} = z^{2 n} $ for two distinct primes $ p $ and $ q $, and to present explicit forms of their solutions $ (p, x, q, y, z, n) $. Recent studies on the exponential Diophantine equation $ p^x+q^y = z^2 $, where $ p $ and $ q $ are primes, have addressed cases where $ p = 2 $ or $ p\equiv q\pmod 4 $. In this paper, we address the case where $ p\not\equiv q\pmod 4 $ and $ y $ is even. In addition, we explore special cases where $ z $ is the prime and provide the complete set of solutions for $ p^x+q^{2y} = z^{2n} $. We also show that the equation has no solution when $ \{2, 3\}\nsubseteq\{p, q, z\} $. In other words, we provide almost explicit solutions to $ p^x+ q^{y} = z^{2 n} $ except for the case where both $ x $ and $ y $ are odd.
| [1] | D. Acu, On a Diophantine equation $2^x+5^y = z^2$, Gen. Math., 15 (2007), 145–148. |
| [2] |
S. Chotchaisthit, On the Diophantine equation $p^x+(p+1)^y = z^2$ where $p$ is a Mersenne prime, IJPAM, 88 (2013), 169–172. http://dx.doi.org/10.12732/ijpam.v88i2.2 doi: 10.12732/ijpam.v88i2.2
|
| [3] |
N. Burshtein, On solutions to the Diophantine equations $p^x+q^y = z^4$, APAM, 14 (2017), 63–68. http://dx.doi.org/10.22457/apam.v14n1a8 doi: 10.22457/apam.v14n1a8
|
| [4] | P. B. Borah, M. Dutta, On the Diophantine equation $7^x + 32^y = z^2$ and its generalization, Integers, 22 (2022), A29. |
| [5] |
Y. Fujita, M. Le, A parametric family of ternary purely exponential Diophantine equation ${A}^x+{B}^y = {C}^z$, Turk. J. Math., 46 (2022), 1224–1232. https://doi.org/10.55730/1300-0098.3153 doi: 10.55730/1300-0098.3153
|
| [6] | W. S. Gayo-Jr., J. B. Bacani, On the solutions of the Diophantine equation ${M}^x+({M}-1)^y = z^2$, Ital. J. Pure Appl. Math., 47 (2022), 1113–1117. |
| [7] |
K. Laipaporn, S. Wananiyakul, P. Khachorncharoenkul, The Diophantine equation $a^x\pm a^y = z^n$ when $a$ is any nonnegative integer, J. Math. Comput. SCI-JM, 32 (2024), 213–221. http://dx.doi.org/10.22436/jmcs.032.03.02 doi: 10.22436/jmcs.032.03.02
|
| [8] |
N. Burshtein, Solutions of the Diophantine equations $2^x+p^y = z^2$ when $p$ is prime, APAM, 16 (2018), 471–477. http://dx.doi.org/10.22457/apam.v16n2a25 doi: 10.22457/apam.v16n2a25
|
| [9] |
R. J. S. Mina, J. B. Bacani, Non-existence of solutions of Diophantine equations of the form $p^x+q^y = z^2n$, Math. Stat., 7 (2019), 78–81. http://dx.doi.org/10.13189/ms.2019.070304 doi: 10.13189/ms.2019.070304
|
| [10] |
R. J. S. Mina, J. B. Bacani, On the solutions of the Diophantine equation $p^x+(p+4k)^y = z^2$ for prime pairs $p$ and $p+4k$, EJPAM, 14 (2021), 471–479. https://doi.org/10.29020/nybg.ejpam.v14i2.3947 doi: 10.29020/nybg.ejpam.v14i2.3947
|
| [11] |
B. Sroysang, On the Diophantine equation $47^x+49^y = z^2$, IJPAM, 89 (2013), 279–282. https://doi.org/10.12732/ijpam.v89i2.11 doi: 10.12732/ijpam.v89i2.11
|
| [12] |
N. Burshtein, On solutions of the Diophantine equations $8^x+9^y = z^2$ when $x, y, z$ are positive integers, APAM, 20 (2019), 79–83. http://dx.doi.org/ 10.22457/apam.641v20n2a6 doi: 10.22457/apam.641v20n2a6
|
| [13] | M. G. Leu, G. W. Li, The Diophantine equation $2x^2+1 = 3^n$, Proc. Amer. Math. Soc., 131 (2003), 3643–3645. |
| [14] | W. Sierpiński, O równaniu $3^x+4^y = 5^z$, Wiadom. Mat., 1 (1956), 194–195. |
| [15] | L. Jeśmanowicz, Kilka uwag o liczbach pitagorejskich, Wiadom. Mat., 1 (1956), 196–202. |
| [16] | D. Acu, On the Diophantine equations of type $a^x+b^y = c^z$, Gen. Math., 13 (2005), 67–72. |
| [17] |
R. Scott, R. Styer, Number of solutions to $a^x + b^y = c^z$, Publ. Math., 88 (2016), 131–138. http://dx.doi.org/10.5486/PMD.2016.7282 doi: 10.5486/PMD.2016.7282
|
| [18] |
H. Yang, R. Fu, A note on Jeśmanowicz' conjecture concerning primitive Pythagorean triples, J. Number Theory, 156 (2015), 183–194. https://doi.org/10.1016/j.jnt.2015.04.009 doi: 10.1016/j.jnt.2015.04.009
|
| [19] |
P. Yuan, Q. Han, Jeśmanowicz' conjecture and related equations, Acta Arith., 184 (2018), 37–49. http://dx.doi.org/10.4064/aa170508-17-9 doi: 10.4064/aa170508-17-9
|
| [20] |
E. Kizildere, M. Le, G. Soydan, A note on the ternary purely exponential diophantine equation $A^x + B^y = C^z$ with $A + B = C^2$, Stud. Sci. Math. Hung., 57 (2020), 200–206. http://dx.doi.org/10.1556/012.2020.57.2.1457 doi: 10.1556/012.2020.57.2.1457
|
| [21] | U. Pintoptang, S. Tadee, The complete set of non-negative integer solutions for the Diophantine equation $(pq)^2x+p^{y} = z^{2}$, where $p, q, x, y, z$ are non-negative integers with $p$ prime and $p \nmid q$, IJMCS, 18 (2023), 205–209. |
| [22] |
K. Laipaporn, S. Kaewchay, A. Karnbanjong, On a Diophantine equations $a^x+b^y+c^z = w^2$, EJPAM, 16 (2023), 2066–2081. https://doi.org/10.29020/nybg.ejpam.v16i4.4936 doi: 10.29020/nybg.ejpam.v16i4.4936
|
| [23] |
S. Fei, G. Zhu, R. Wu, On a conjecture concerning the exponential Diophantine equation $(an^2+1)^x+(bn^2-1)^y = (cn)^z$, Electronic Res. Arch., 32 (2024), 4096–4107. http://dx.doi.org/10.3934/era.202418 doi: 10.3934/era.202418
|
| [24] |
C. Panraksa, Exploring $8^x + n^y = z^2$ through associated elliptic curves, IJMCS, 20 (2025), 247–254. https://doi.org/10.69793/ijmcs/01.2025/chatchawan doi: 10.69793/ijmcs/01.2025/chatchawan
|
| [25] |
R. Scott, R. Styer, On $p^x-q^y = c$ and related three term exponential Diophantine equations with primes bases, J. Number Theory, 105 (2004), 212–234. http://dx.doi.org/10.1016/j.jnt.2003.11.008 doi: 10.1016/j.jnt.2003.11.008
|
| [26] | N. Viriyapong, C. Viriyapong, On the Diophantine equation $n^x+19^y = z^2$, where $n \equiv 2 \pmod 57$, IJMCS, 17 (2022), 1639–1642. |
| [27] |
N. Burshtein, On solutions to the Diophantine equations $5^x+103^y = z^2$ and $5^x+11^y = z^2$ with positive integers $x, y, z$, APAM, 19 (2019), 75–77. http://dx.doi.org/10.22457/apam.607v19n1a9 doi: 10.22457/apam.607v19n1a9
|
Kittipong Laipaporn, Saeree Wananiyakul, Prathomjit Khachorncharoenkul. Explicit solutions and non-solutions for the Diophantine equation $ p^x+ q^{2y} = z^{2 n} $ involving primes $ p\not\equiv q \pmod 4 $[J]. AIMS Mathematics, 2025, 10(7): 15720-15736. doi: 10.3934/math.2025704
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