On the Diophantine Equations and , Where and are Primes
Abstract
In this paper, we study Diophantine equations and , where and are primes. We found that all non-negative integer solutions of the Diophantine equation are of the following and all non-negative integer solutions of the Diophantine equationare of the following. Keywords : Diophantine equation ; Catalan’s ConjectureReferences
Bacani, J. B., & Rabago, J.F.T. (2015). The complete set of solutions of the Diophantine equation p^x+q^y=z^2 for twin primes p and q . International Journal of Pure and Applied Mathematics, 104, 517-521.
Burshtein, N. (2017). On solutions to the Diophantine equation p^x+q^y=z^4 , Annals of Pure and Applied Mathematics. 14, 63-68.
Burshtein, N. (2018). On the Diophantine equation 2^(2x+1)+7^y=z^2 . Annals of Pure and Applied Mathematics, 16, 177-179.
Burshtein, N. (2019). All the solutions of the Diophantine equations p^x+p^y=z^2 and p^x-p^y=z^2 when p>=2 is prime. Annals of Pure and Applied Mathematics, 19, 111-119.
Burshtein, N. (2020). All the solutions of the Diophantine equations p^x+p^y=z^4 when p>=2 is prime and x,y,z are positive integers. Annals of Pure and Applied Mathematics, 21, 125-128.
Burshtein, N. (2021). All the solutions of the Diophantine equations p^4+q^y=z^4 and p^4-q^y=z^4 when p,q are distinct primes. Annals of Pure and Applied Mathematics, 23, 17-20.
Chotchaisthit, S. (2012). On the Diophantine equation 4^x+p^y=z^2 where p is a prime number. Amer. J. Math. Sci., 1, 191-193.
Chotchaisthit, S. (2013). On the Diophantine equation 2^x+11^y=z^2 . Maejo International Journal of Science and
Technology, 7, 291-293.
Chotchaisthit, S. (2013). On the Diophantine equation p^x+(p+1)^y=z^2 where p is a Mersenne prime. International Journal of Pure and Applied Mathematics, 88, 169-172.
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, 179-183.
Mihailescu, P. (2004). Primary cyclotomic units and a proof of Catalan’s conjecture. J. Reine Angew. Math., 572,167-195.
Mina, R. J. S., & Bacani, J.B. (2019). Non-existence of solutions of Diophantine equations of the form p^x+q^y=z^2n. Mathematics and Statistics, 7, 78-81.
Mina, R. J. S., & Bacani, J.B. (2021). On the solutions of the Diophantine equation p^x+(p+4k)^y=z^2 for prime pairs p and p+4k . European Journal of Pure and Applied Mathematics, 14, 471-479.
Singha, B. (2021). Non-negative solutions of the nonlinear Diophantine equation (8^n)^x+p^y=z^2 for some prime number p . Walailak J. Sci. & Tech., 18:11719, 8 pages.
Sroysang, B. (2012). On the Diophantine equation 31^x+32^y=z^2 . International Journal of Pure and Applied Mathematics, 81, 609-612.
Suvarnamani, A., Singta, A., & Chotchaisthit, S. (2011). On two Diophantine equations 4^x+7^y=z^2 and 4^x+11^y=z^2. Sci. Techno. RMUTT J, 1, 25-28.
Burshtein, N. (2017). On solutions to the Diophantine equation p^x+q^y=z^4 , Annals of Pure and Applied Mathematics. 14, 63-68.
Burshtein, N. (2018). On the Diophantine equation 2^(2x+1)+7^y=z^2 . Annals of Pure and Applied Mathematics, 16, 177-179.
Burshtein, N. (2019). All the solutions of the Diophantine equations p^x+p^y=z^2 and p^x-p^y=z^2 when p>=2 is prime. Annals of Pure and Applied Mathematics, 19, 111-119.
Burshtein, N. (2020). All the solutions of the Diophantine equations p^x+p^y=z^4 when p>=2 is prime and x,y,z are positive integers. Annals of Pure and Applied Mathematics, 21, 125-128.
Burshtein, N. (2021). All the solutions of the Diophantine equations p^4+q^y=z^4 and p^4-q^y=z^4 when p,q are distinct primes. Annals of Pure and Applied Mathematics, 23, 17-20.
Chotchaisthit, S. (2012). On the Diophantine equation 4^x+p^y=z^2 where p is a prime number. Amer. J. Math. Sci., 1, 191-193.
Chotchaisthit, S. (2013). On the Diophantine equation 2^x+11^y=z^2 . Maejo International Journal of Science and
Technology, 7, 291-293.
Chotchaisthit, S. (2013). On the Diophantine equation p^x+(p+1)^y=z^2 where p is a Mersenne prime. International Journal of Pure and Applied Mathematics, 88, 169-172.
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, 179-183.
Mihailescu, P. (2004). Primary cyclotomic units and a proof of Catalan’s conjecture. J. Reine Angew. Math., 572,167-195.
Mina, R. J. S., & Bacani, J.B. (2019). Non-existence of solutions of Diophantine equations of the form p^x+q^y=z^2n. Mathematics and Statistics, 7, 78-81.
Mina, R. J. S., & Bacani, J.B. (2021). On the solutions of the Diophantine equation p^x+(p+4k)^y=z^2 for prime pairs p and p+4k . European Journal of Pure and Applied Mathematics, 14, 471-479.
Singha, B. (2021). Non-negative solutions of the nonlinear Diophantine equation (8^n)^x+p^y=z^2 for some prime number p . Walailak J. Sci. & Tech., 18:11719, 8 pages.
Sroysang, B. (2012). On the Diophantine equation 31^x+32^y=z^2 . International Journal of Pure and Applied Mathematics, 81, 609-612.
Suvarnamani, A., Singta, A., & Chotchaisthit, S. (2011). On two Diophantine equations 4^x+7^y=z^2 and 4^x+11^y=z^2. Sci. Techno. RMUTT J, 1, 25-28.
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Published
2023-01-05
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