On the Diophantine Equation 3^x+n^y=z^3
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Abstract
By using elementary methods and Mihailescu’s Theorem, the non-existence of integer solutions for the title Diophantine equation is investigated, where n is a positive integer and x, y, z are non-negative integers with some conditions.
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References
Sroysang, B. (2014). On the Diophantine equation 3^x+45^y=z^2. International Journal of Pure and Applied Mathematics. 91(2), 269-272.
Qi, L. and Li, X. (2015). The Diophantine equation 8^x+p^y=z^2. The Scientific World Journal. 2015, Article ID 306590, 3 pages.
Asthana, S. and Singh, M. M. (2017). On the Diophantine equation 3^x+13^y=z^2. International Journal of Pure and Applied Mathematics. 114(2), 301-304.
Orosram, W. and Comemuang, C. (2020). On the Diophantine equation 8^x+n^y=z^2. WSEAS Transactions on Mathematics. 19, 520-522.
Tangjai, W. and Chubthaisong, C. (2021). On the Diophantine equation 3^x+p^y=z^2 where p≡2 (mod 3). WSEAS Transactions on Mathematics. 20, 283-287.
Viriyapong, N. and Viriyapong, C. (2021). On the Diophantine equation n^x+13^y=z^2 where n≡2 (mod 39) and n+1 is not a square number. WSEAS Transactions on Mathematics. 20, 442-445.
Borah, P. B. and Dutta, M. (2022). On the Diophantine equation 7^x+32^y=z^2 and its generalization. Integers. 22, 1-5.
Tadee, S. (2023). On the Diophantine equation n^x+10^y=z^2. WSEAS Transactions on Mathematics. 22, 150-153.
Tadee, S. and Siraworakun, A. (2023). Non-existence of positive integer solutions of the Diophantine equation p^x+(p+2q)^y=z^2, where p,q and p+2q are prime numbers. European Journal of Pure and Applied Mathematics. 16(2), 724-735.
Burshtein, N. (2017). All the solutions of the Diophantine equation p^3+q^2=z^3. Annals of Pure and Applied Mathematics. 14(2), 207-211.
Burshtein, N.(2018).The infinitude of solutions to the Diophantine equation p^3+q=z^3 when p,q are primes. Annals of Pure and Applied Mathematics. 17(1), 135-136.
Burshtein, N. (2020). On solutions to the Diophantine equations p^x+q^y=z^3 when p≥2,q are primes and 1≤x,y≤2 are integers. Annals of Pure and Applied Mathematics. 22(1), 13-19.
Burshtein, N. (2021). All the solutions of the Diophantine equation p^3+q^y=z^3 with distinct odd primes p,q when y>3. Journal of Mathematics and Informatics. 20, 1-4.
Tadee, S. (2023). The Diophantine equations 8^x+p^y=z^3 and 8^x-p^y=z^3. Science and Technology Nakhon Sawan Rajabhat University Journal. 15(21), 98-106.
Mihailescu, P. (2004). Primary cyclotomic units and a proof of Catalan’s conjecture. Journal für die Reine und Angewandte Mathematik. 572, 167-195.