On the Diophantine Equation t^x + (t + 3k)^y = z^2 Where t is a Positive Integer

Authors

  • Darakorn Jantoy Faculty of Science, Burapha University
  • Somkid Intep Faculty of Science, Burapha University

Keywords:

Diophantine equation , integer solution , non-negative integer

Abstract

Background and Objectives : to find the solution to the Diophantine equation tx+ (t + 3k)y = z2  where  k, x, y  and   z   are non-negative integers, and  is a positive integer, which is in the form 3n+1 for n  some non-negative integer .

Methodology : proving by contradiction and various properties related to the congruent in order to find the Diophantine equation’s solutions.

Main Results : the Diophantine equation    tx+ (t + 3k)y = z2   has no any solution.

Conclusions  : the Diophantine equation tx+ (t + 3k)y = z2   where  t    is a positive integer, which is in the form 3n+1  for some non-negative integer  n , has no any solution for k, x, y  and z are non-negative integers.

References

Asthana, S., & Singh, M. (2017). On the Diophantine Equation 3^x+13^y=z^2 , International Journal of Pure and Applied Mathematics, 114(2), 301-304.

Burshtein, N. (2018). Solution of the Diophantine Equation p^x+(p+6)^y=z^2 when p,(p+6) are Primes and x+y=2,3,4 . Annals of Pure and Applied Mathematics, 18(1), 101-106.

Chotchaisthit, S. (2012). On the Diophantine Equation 4^x+p^y=z^2 where p is a Prime Number. American Journal Mathematics and Science, 1(1), 191-193.

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.

Oliveria, 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.

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Published

2024-04-11

How to Cite

Jantoy, D., & Intep, S. (2024). On the Diophantine Equation t^x + (t + 3k)^y = z^2 Where t is a Positive Integer. Burapha Science Journal, 29(1), 402–407. Retrieved from https://li05.tci-thaijo.org/index.php/buuscij/article/view/241

Issue

Vol. 29 No. 1 (2024): Burapha Science Journal

Section

Research Articles

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