Solution of the Diophantine equation 22x+40y=z2

Sudhanshu Aggarwal; Lalit Mohan Upadhyaya

doi:10.48165/bpas.2023.42E.2.3

Authors

Sudhanshu Aggarwal Department of Mathematics, National Post Graduate College, Barhalganj, Gorakhpur, Uttar Pradesh-273402, India.
Lalit Mohan Upadhyaya Department of Mathematics, Municipal Post Graduate College, Mussoorie, Dehradun, Uttarakhand -248179, India.

DOI:

https://doi.org/10.48165/bpas.2023.42E.2.3

Keywords:

Catalan’s Conjecture, Diophantine Equation, Solution

Abstract

Since Diophantine equations play an important role in solving important real-world problems such as business investment problems, network flow problems, pole placement problems, and data privacy problems, researchers are increasingly interested in developing new techniques for analyzing the nature and solutions of the various Dio phantine equations. In this study we investigate the Diophantine problem 22x+40y = z2, where x, y, z are non-negative integers, and discover that it does not have a non-negative integer solution.

References

Koshy, T. (2007). Elementary Number Theory with Applications, Second Edition, Academic Press, USA.

Aggarwal, S., Sharma, S. D. and Singhal, H. (2020). On the Diophantine equation 223x+241y = z2, 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 181x+199y = z2, International Journal of Latest Technology in Engineering, Management & Applied Science, 9 (8), 85–86.

Aggarwal, S. and Sharma, N. (2020). On the non-linear Diophantine equation 379x + 397y = z2, Open Journal of Mathematical Sciences, 4(1), 397–399.

Aggarwal, S. (2020). On the existence of solution of Diophantine equation 193x + 211y = z2, 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 (132m)+(6r + 1)n = z2, Journal of Scientific Research, 13(3), 845–849.

Aggarwal, S. and Upadhyaya, L. M. (2022). On the Diophantine equation 8α + 67β = γ2, Bull. Pure Appl. Sci. Sect. E Math. Stat., 41(2), 153–155.

Goel, P., Bhatnagar, K. and Aggarwal, S. (2020). On the exponential Diophantine equation M5p+ M7q = r2, International Journal of Interdisciplinary Global Studies, 14(4), 170–171. [9] Bhatnagar, K. and Aggarwal, S. (2020). On the exponential Diophantine equation 421p+439q = r2, International Journal of Interdisciplinary Global Studies, 14(4), 128–129.

Gupta, D., Kumar, S. and Aggarwal, S. (2022). Solution of non-linear exponential Diophantine equation (xa + 1)m +(yb + 1)n= z2, Journal of Emerging Technologies and Innovative Research, 9(9), f154-f157.

Gupta, D., Kumar, S. and Aggarwal, S. (2022). Solution of non-linear exponential Diophantine equation xα + (1 + my)β = z2, Journal of Emerging Technologies and Innovative Research, 9(9), d486-d489.

Hoque, A. and Kalita, H. (2015). On the Diophantine equation (pq − 1)x + pqy = z2, Journal of Analysis & Number Theory, 3(2), 117–119.

Kumar, A., Chaudhary, L. and Aggarwal, S. (2020). On the exponential Diophantine equation 601p + 619q = r2, 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 (22m+1 − 1)+(6r+1 + 1)n = ω2, International Journal of Interdisciplinary Global Studies, 14(4), 183–184.

Kumar, S., Bhatnagar, K., Kumar, N. and Aggarwal, S. (2020). On the exponential Diophantine equation (72m)+ (6r + 1)n = z2, International Journal of Interdisciplinary Global Studies, 14(4), 181–182.

Mishra, R., Aggarwal, S. and Kumar, A. (2020). On the existence of solution of Diophantine equation 211α + 229β = γ2, International Journal of Interdisciplinary Global Studies, 14(4), 78– 79.

Schoof, R. (2008) Catalan’s Conjecture, Springer-Verlag, London.

Sroysang, B. (2014). On the Diophantine equation 323x + 325y = z2, International Journal of Pure and Applied Mathematics, 91(3), 395–398.

Sroysang, B. (2014). On the Diophantine equation 3x + 45y = z2, International Journal of Pure and Applied Mathematics, 91(2), 269–272.

Sroysang, B. (2014). On the Diophantine equation143x + 145y = z2, International Journal of Pure and Applied Mathematics, 91(2), 265–268.

Sroysang, B. (2014). On the Diophantine equation 3x + 85y = z2, International Journal of Pure and Applied Mathematics, 91(1), 131–134.

Sroysang, B. (2014). More on the Diophantine equation 4x + 10y = z2, International Journal of Pure and Applied Mathematics, 91(1), 135–138.

Aggarwal, S., Swarup, C., Gupta, D. and Kumar, S. (2022). Solution of the Diophantine equation 143x + 45y = z2, Journal of Advanced Research in Applied Mathematics and Statistics, 7(3 & 4), 1–4.

Aggarwal, S., Kumar, S., Gupta, D. and Kumar, S. (2023). Solution of the Diophantine equation 143x + 485y = z2, International Research Journal of Modernization in Engineering Technology and Science, 5(2), 555–558.

Aggarwal, S., Swarup, C., Gupta, D. and Kumar, S. (2023). Solution of the Diophantine equation 143x + 85y = z2, International Journal of Progressive Research in Science and Engineering, 4(2), 5–7.

Aggarwal, S., Shahida A. T., Pandey, E. and Vyas, A. (2023). On the problem of solution of non-linear (exponential) Diophantine equation βx + (β + 18)y = z2, Mathematics and Statistics, 11(5), 834–839.