W. Nilsrakoo - Department of Mathematics, Statistics and Computer, Faculty of Science, Ubon Ratchathani University, Ubon Ratchathani, 34190, Thailand. T. Wichana - Department of Mathematics, Statistics and Computer, Faculty of Science, Ubon Ratchathani University, Ubon Ratchathani, 34190, Thailand. A. Nilsrakoo - Department of Mathematics, Faculty of Science, Ubon Ratchathani Rajabhat University, Ubon Ratchathani, 34000, Thailand.
The purpose of this paper is to study non-negative integer solutions \((x,y,z)\) of the Diophantine equation \(p^x+(2p)^y=z^2\), where \(p\) is a prime number, using elementary techniques and Catalan's conjecture to prove. Results are obtained that the equation has non-negative integer solutions if \(p=2\) or \(p\) is a Fermat prime or a Mersenne prime; the equation has a unique non-negative integer solution if \(p\) is a Fermat prime or a Mersenne prime with \(p>3\); the equation has no non-negative integer solutions if \(p\not\equiv 1\pmod{8}\) or \(p\equiv 3,5,6\pmod{7}\) or \(\ord{p}{(2)}\neq 2m\) for every odd positive integer \(m\); all non-negative integer solutions \((x,y,z)\) are in certain form whenever \(p\) is a prime number such that \(p\equiv 1,9,25\pmod{56}\) with \(\ord{p}{(2)}=2m\) for some odd positive integer \(m>1\).
W. Nilsrakoo, T. Wichana, A. Nilsrakoo, On the Diophantine equation \(p^x+(2p)^y=z^2\), where \(p\) is prime, Journal of Mathematics and Computer Science, 41 (2026), no. 4, 571--580
Nilsrakoo W., Wichana T., Nilsrakoo A., On the Diophantine equation \(p^x+(2p)^y=z^2\), where \(p\) is prime. J Math Comput SCI-JM. (2026); 41(4):571--580
Nilsrakoo, W., Wichana, T., Nilsrakoo, A.. "On the Diophantine equation \(p^x+(2p)^y=z^2\), where \(p\) is prime." Journal of Mathematics and Computer Science, 41, no. 4 (2026): 571--580
