Solution of the Diophantine Equation 323x + 85y = z2

Sudhanshu Aggarwal

Sudhanshu Aggarwal Department of Mathematics, National Post Graduate College, Barhalganj, Gorakhpur, Uttar Pradesh, India.

Abstract

Diophantine equations have numerous applications in Algebra, Chemistry, Astrology, Cryptography and Trigonometry. In this paper, author examined the Diophantine equation 323x + 85 =z2, where x, y, z are non-negative integers, for its non-negative integer solutions. For this, author used Catalan’s conjecture and proved that (x, y, z) = (1, 0, 18) is the unique non-negative integer solution of the Diophantine equation 323x + 85 =z2, where x, y, z are non-negative integers.

References

1. Koshy, T. (2007) Elementary Number Theory with Applications, Second Edition, Academic Press, USA.
2. Aggarwal, S., Sharma, S.D. and Singhal, H. (2020) On the Diophantine equation 223^x+241^y=z^2, International Journal of Research and Innovation in Applied Science, 5 (8), 155-156.
3. Aggarwal, S., Sharma, S.D. and Vyas, A. (2020) On the existence of solution of Diophantine equation 181^x+199^y=z^2, International Journal of Latest Technology in Engineering, Management & Applied Science, 9 (8), 85-86.
4. Aggarwal, S. and Sharma, N. (2020) On the non-linear Diophantine equation379^x+397^y=z^2, Open Journal of Mathematical Sciences, 4(1), 397-399.
5. Aggarwal, S. (2020) On the existence of solution of Diophantine equation 193^x+211^y=z^2, Journal of Advanced Research in Applied Mathematics and Statistics, 5(3 & 4), 4-5.
6. Aggarwal, S. and Kumar, S. (2021) On the exponential Diophantine equation (13^2m )+(6r+1)^n=z^2, Journal of Scientific Research, 13(3), 845-849.
7. Aggarwal, S. and Upadhyaya, L.M. (2022) On the Diophantine equation 8^α+67^β=γ^2, Bulletin of Pure & Applied Sciences-Mathematics and Statistics, 41(2), 153-155.
8. Goel, P., Bhatnagar, K. and Aggarwal, S. (2020) On the exponential Diophantine equation 〖M_5〗^p+〖M_7〗^q=r^2, International Journal of Interdisciplinary Global Studies, 14(4), 170-171.
9. Bhatnagar, K. and Aggarwal, S. (2020) On the exponential Diophantine equation 421^p+439^q=r^2, International Journal of Interdisciplinary Global Studies, 14(4), 128-129.
10. Gupta, D., Kumar, S. and Aggarwal, S. (2022) Solution of non-linear exponential Diophantine equation (x^a+1)^m+(y^b+1)^n=z^2, Journal of Emerging Technologies and Innovative Research, 9(9), f154-f157.
11. Gupta, D., Kumar, S. and Aggarwal, S. (2022) Solution of non-linear exponential Diophantine equation x^α+(1+my)^β=z^2, Journal of Emerging Technologies and Innovative Research, 9(9), d486-d489.
12. Hoque, A. and Kalita, H. (2015) On the Diophantine equation (p^q-1)^x+p^qy=z^2, Journal of Analysis & Number Theory, 3(2), 117-119.
13. Kumar, A., Chaudhary, L. and Aggarwal, S. (2020) On the exponential Diophantine equation 601^p+619^q=r^2, International Journal of Interdisciplinary Global Studies, 14(4), 29-30.
14. Kumar, S., Bhatnagar, K., Kumar, A. and Aggarwal, S. (2020) On the exponential Diophantine equation (2^(2m+1)-1)+〖(6^(r+1)+1)〗^n=ω^2, International Journal of Interdisciplinary Global Studies, 14(4), 183-184.
15. Kumar, S., Bhatnagar, K., Kumar, N. and Aggarwal, S. (2020) On the exponential Diophantine equation (7^2m )+〖(6r+1)〗^n=z^2, International Journal of Interdisciplinary Global Studies, 14(4), 181-182.
16. Mishra, R., Aggarwal, S. And Kumar, A. (2020) On the existence of solution of Diophantine equation 211^α+229^β=γ^2, International Journal of Interdisciplinary Global Studies, 14(4), 78-79.
17. Schoof, R. (2008) Catalan’s conjecture, Springer-Verlag, London.
18. Sroysang, B. (2014) On the Diophantine equation 323^x+325^y=z^2, International Journal of Pure and Applied Mathematics, 91(3), 395-398.
19. Sroysang, B. (2014) On the Diophantine equation 3^x+45^y=z^2, International Journal of Pure and Applied Mathematics, 91(2), 269-272.
20. Sroysang, B. (2014) On the Diophantine equation143^x+145^y=z^2, International Journal of Pure and Applied Mathematics, 91(2), 265-268.
21. Sroysang, B. (2014) On the Diophantine equation 3^x+85^y=z^2, International Journal of Pure and Applied Mathematics, 91(1), 131-134.
22. Sroysang, B. (2014) More on the Diophantine equation 4^x+10^y=z^2, International Journal of Pure and Applied Mathematics, 91(1), 135-138.
23. Aggarwal, S., Swarup, C., Gupta, D. and Kumar, S. (2022) Solution of the Diophantine equation 143^x+45^y=z^2, Journal of Advanced Research in Applied Mathematics and Statistics, 7(3 & 4), 1-4.
24. Aggarwal, S., Kumar, S., Gupta, D. and Kumar, S. (2023) Solution of the Diophantine equation143^x+485^y=z^2, International Research Journal of Modernization in Engineering Technology and Science, 5(2), 555-558.
25. Aggarwal, S., Swarup, C., Gupta, D. and Kumar, S. (2023) Solution of the Diophantine equation 143^x+85^y=z^2, International Journal of Progressive Research in Science and Engineering, 4(2), 5-7.