On the Diophantine Equation 13^x+a^y=z^3
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Abstract
In this article, we study the non-negative integer solutions of the Diophantine equation , where
is a positive integer and
are non-negative integers, by using the basic concepts of congruence and Mihăilescu’s Theorem. These findings indicate that if
, then the Diophantine equation has no non-negative integer solution. Moreover, the Diophantine equation has the non-negative integer solution
, where
and
is an integer.
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References
Burshtein, N. (2017). All the Solutions of the Diophantine Equation p3 + q2 = z3. Annals of Pure and Applied Mathematics, 14(2), 207-211. http://dx.doi.org/10.22457/apam.v14n2a1
Burshtein, N. (2018). The Infinitude of Solutions to the Diophantine Equation p3 + q = z3 when p, q are Primes. Annals of Pure and Applied Mathematics, 17(1), 135-136. http://dx.doi.org/10.22457/apam.v17n1a15
Burshtein, N. (2020). On Solutions to the Diophantine Equations px + qy = z3 when p2, q are Primes and 1x, y2 are Integers. Annals of Pure and Applied Mathematics, 22(1), 13-19. http://dx.doi.org/10.22457/apam.v22n1a03679
Burshtein, N. (2021). All the Solutions of the Diophantine Equation p3 + qy = z3 with Distinct Odd Primes p, q when y > 3. Journal of Mathematics and Informatics, 20(1), 1-4. http://dx.doi.org/10.22457/jmi.v20a01188
Mihăilescu, P. (2004). Primary Cyclotomic Units and a Proof of Catalan’s Conjecture. Journal für die Reine und Angewandte Mathematik, 2004(572), 167-195. http://dx.doi.org/10.1515/crll.2004.048
Mina, R.J.S. and Bacani, J.B. (2022). On a Diophantine Equation of Type px + qy = z3. Konuralp Journal of Mathematics, 10(1), 55-58.
Niven, I., Zuckerman, H.S. and Montgomery, H.L. (1991). An Introduction to the Theory of Numbers (5thed). New York: John Wiley & Sons, Inc.
Tadee, S. (2023). The Diophantine Equations 8x + py = z3 and 8x - py = z3. Science and Technology Nakhon Sawan Rajabhat University Journal, 15(21), 98-106.
Tadee, S. (2024). On the Diophantine Equation 3x + ny = z3. Rattanakosin Journal of Science and Technology, 6(2), 79-84. https://ph02.tci-thaijo.org/index.php/RJST/article/view/251748