On the Diophantine Equation p^x+(p+7)^y=z^2

Suton Tadee

Authors

Suton Tadee Department of Mathematics, Faculty of Science and Technology, Thepsatri Rajabhat University, Thailand

Keywords:

Diophantine equation, non-negative integer solution, prime number, congruence, Legendre symbol

Abstract

Background and Objectives: One interesting part of number theory is Diophantine analysis, which is the study of integer solutions to equations. Diophantine analysis investigates to answer whether a certain Diophantine equation is unsolvable, or solvable with the unique integer solution, finitely many integer solutions or infinitely many integer solutions. The Diophantine equations come in many forms. One form that has received attention is the exponential Diophantine equation of the form $a^{x}+b^{y}=z^{2}$ , where $a,b$ are fixed positive integers and $x,y,z$ are non-negative integer variables. To find the non-negative integer solutions of the Diophantine equation, the Catalan’s conjecture plays an important role. This Conjecture stated that $\left(a,b,x,z \right)=\left(3,2,2,3 \right)$ is the unique non-negative integer solution of the Diophantine equation $a^{x}-b^{y}=1$ , when $a,b,x$ and $z$ are non-negative integers such that $min\left\{a,b,x,y\right\}> 1$ , which remained unproven for over 150 years until it was proved by Mihăilescu in 2004. Over the past decade, the Diophantine equation of the form $p^{x}+\left(p+n\right)^{y}=z^{2}$ , where $n$ is a fixed positive integer, $p$ is a prime and $x,y,z$ are non-negative integers, has received considerable attention of many mathematicians in the study and search for solutions. In this paper, we solve the Diophantine equation for the case $n=7$ , which has never been studied before.

Methodology: The research and solutions presented here utilize mathematical proofs and fundamental concepts of number theory, such as prime numbers, congruences, and Legendre symbol. Here, some properties of prime number are applied to prove the theorems, such as: Let $p$ be a prime number with $p\notin \left\{2,7\right\}$ . If the Diophantine equation $p^{x}+\left(p+7\right)^{y}=z^2$ has a non-negative integer solution $\left(x,y,z \right)$ and $y$ is even, then $2\left(p+7 \right)^{y/2}=p^{x}-1$ . This will be useful to prove our main results. For the congruent method and Legendre symbol, we often use them in proofs to find contradictions.

Main Results: In this article, by using elementary methods, the Diophantine equation $p^{x}+\left(p+7\right)^{y}=z^2$ , where $p$ is a prime number and $x,y,z$ are non-negative integers, is investigated. The research results found that if $p=2$ , then the Diophantine equation has exactly two non-negative integer solutions. The solutions $\left(x,y,z \right)$ are $\left(3,0,3 \right)$ and $\left(4,1,5 \right)$ . Unfortunately, we are not yet able to find all non-negative integer solutions of the Diophantine equation for the case $p=3$ , but we can show the explicit non-negative integer solutions of the Diophantine equation under the given conditions. If $p=3$ and $y\equiv 0\left(mod4\right)$ , then the Diophantine equation has the unique non-negative integer solution. The solution $\left(x,y,z \right)$ is $\left(1,0,2 \right)$ . For the case $p=5$ , we prove that the Diophantine equation has a unique non-negative integer solution, $\left(x,y,z\right)=\left(2,2,13\right)$ . For the case $p=7$ , we prove that the Diophantine equation has the unique non-negative integer solution, which is $\left(x,y,z\right)=\left(4,5,735\right)$ . For the case $p\equiv 5,11,13,43,45,51\left(mod56\right)$ , if the Diophantine equation has a non-negative integer solution $\left(x,y,z \right)$ , then there exists a non-negative integer $t$ such that $y=2t$ and $2\left(p+7 \right)^{t}=p^{x}-1$ . Moreover, we present some conditions for non-existence of the non-negative integer solutions to the Diophantine equation, namely, (1) if $p\equiv 15\left(mod28\right)$ , then the Diophantine equation has no non-negative integer solution. (2) if $p\equiv 5,11,13,43,45,51\left(mod56\right)$ and there exists a positive integer $n>1$ such that $n$ is not power of $2$ and $p\equiv 1\left(mod\,n\right)$ , then the Diophantine equation has also no non-negative integer solution.

Conclusions: By using elementary properties of prime numbers, congruences, and Legendre symbol, we present explicit non-negative integer solutions $\left(x,y,z \right)$ of the Diophantine equation $p^{x}+\left(p+7\right)^{y}=z^2$ for some prime number $p$ . Note that $p+7$ doesn’t necessarily have to be a prime number. Moreover, we show some conditions of non-existence of non-negative integer solutions to the Diophantine equation. In order to prove our main results, Mihăilescu’s theorem was also applied. This method can be applied to find non-negative integer solutions to similar Diophantine equations. Furthermore, it was found that there are some cases that remain unanswered, such as: case $p\equiv 1\left(mod 8 \right)$ , which is an interesting open problem for further study and research.

References

Burshtein, N. (2018). The Diophantine equation p^x+(p+4)^y=z^2 when p>3 , p+4 are primes is insolvable in positive integers x,y,z. Annals of Pure and Applied Mathematics, 16(2), 283-286.

Burton, D.M. (2010). Elementary number theory. 7th ed. New York: McGraw-Hill.

Dokchan, R. & Pakapongpun, A. (2021). On the Diophantine equation p^x+(p+20)^y=z^2, where p and p+20 are primes. International Journal of Mathematics and Computer Science, 16(1), 179-183.

Gupta, S., Kumar, S. & Kishan, H. (2018). On the non-linear Diophantine equation p^x+(p+6)^y=z^2. Annals of Pure and Applied Mathematics, 18(1), 125-128.

Jaimano, K., Duanthip, N., Phrommarat, N. & Butsan, T. (2025). On the Diophantine equations p^x+(p+30)^y=z^2. Journal of Science and Technology CRRU, 4(2), 21-26. (in Thai)

Kumar, S., Gupta, D. & Kishan, H. (2019). On the Solutions of exponential Diophantine equation p^x+(p+12)^y=z^2. International Transactions in Mathematical Sciences and Computers, 11(1),1-19.

Leu, M. & Li, G. (2003). The Diophantine equation 2x^2+1=3^n. Proceedings of the American Mathematical Society, 131(12), 3643-3645.

Mihăilescu, P. (2004). Primary cyclotomic units and a proof of Catalan’s conjecture. Journal für die Reine und Angewandte Mathematik, 572, 167-195.

Mina, R.J.S. & Bacani, J.B. (2022). On a Diophantine equation of type p^x+q^y=z^3. Konuralp Journal of Mathematics, 10(1), 55-58.

Mina, R.J.S. & Bacani, J.B. (2023). On the Diophantine equation p^x+(p+5)^y=z^2, where p is odd prime. Thai Journal of Mathematics, 21(1), 67-75.

Nilsrakoo, W., Wichana, T., & Nilsrakoo, A. (2026). On the Diophantine equation p^x+(2p)^y=z^2, where p is prime. Journal of Mathematics and Computer Science, 41, 571-580.

Oliveira, F.N. (2018). On the solvability of the Diophantine equation p^x+(p+8)^y=z^2 when p>3 and p+8 are primes. Annals of Pure and Applied Mathematics, 18(1), 9-13.

Orosram, W. (2023). On the Diophantine equation (p+n)^x+p^y=z^2 where p and p+n are prime numbers. European Journal of Pure and Applied Mathematics, 16(4), 2208-2212.

Pandichelvi, V. & Vanaja, R. (2022). Inspecting integer solutions for an exponential Diophantine equation p^x+(p+2)^y=z^2. Advances and Applications in Mathematical Sciences, 21(8), 4693-4701.

Samaapat, N., Pomsiri, C., & Tadee, S. (2025). On the Diophantine equation 3^x+n^y=z^2, where n≡2 (mod 3) . Journal of Science and Technology CRRU, 4(2), 1-6. (in Thai)

Tadee, S. (2022a). On the Diophantine equation p^x+(p+10)^y=z^2 when p and p+10 are primes. Udon Thani Rajabhat University Journal of Sciences and Technology, 10(2), 155-162. (in Thai)

Tadee, S. (2022b). On the Diophantine equation p^x+(p+14)^y=z^2 where p,p+14 are primes. Annals of Pure and Applied Mathematics, 26(2), 125-130.

Vesarachasart, S., Eiemrawd, C., Thiengtham, S. & Mungkun, A. (2023). On the Diophantine equation p^x+(p+12)^y=z^2 where p and p+12 are prime numbers. Thai Journal of Science and Technology, 12(1), 514-521. (in Thai)

Wannaphan, C. & Tadee, S. (2024). On the Diophantine equation p^x+(p+1)^y=z^2,where p is prime. Journal of Science Ladkrabang, 33(1), 1-7. (in Thai)