The Study of the Exponential Diophantine Equation (ðŸ‘ðŸ•)ðŸð’™+(ðŸ”ð’“+ðŸ)ð’š=ð’›ðŸ
Abstract
Discussing the solution of the Diophantine equation (37)2ð‘¥+(6ð‘Ÿ+1)ð‘¦=ð‘§2, where ð‘Ÿ,ð‘¥,ð‘¦, and 𑧠are non-negative integers, is the goal of this study. The authors employed the congruence modulo approach for this. Findings of the study indicate that the Diophantine equation (37)2ð‘¥+(6ð‘Ÿ+1)ð‘¦=ð‘§2, where ð‘Ÿ,ð‘¥,ð‘¦, and 𑧠are non-negative integers, has no solution in the set of non-negative integers.
References
1. Thomas Koshy (2007) Elementary number theory with applications, 2nd edition, Academic Press, Amsterdam; Boston.
2. Singh, K. (2020) Number theory step by step, Oxford University Press, United Kingdom.
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4. Aggarwal, S., Sharma, S.D. and Singhal, H. (2020). On the Diophantine equation 223ð‘¥+241ð‘¦=ð‘§2, International Journal of Research and Innovation in Applied Science, 5 (8), 155-156.
5. Aggarwal, S., Sharma, S.D. and Vyas, A. (2020). On the existence of solution of Diophantine equation 181ð‘¥+199ð‘¦=ð‘§2, International Journal of Latest Technology in Engineering, Management & Applied Science, 9 (8), 85-86.
6. Kumar, A., Chaudhary, L. and Aggarwal, S. (2020). On the exponential Diophantine equation 601ð‘+619ð‘ž=ð‘Ÿ2, International Journal of Interdisciplinary Global Studies, 14(4), 29-30.
7. Sroysang, B. (2012) More on the Diophantine equation 8ð‘¥+19ð‘¦=ð‘§2, International Journal of Pure and Applied Mathematics, 81(4), 601-604.
8. Mishra, R., Aggarwal, S. and Kumar, A. (2020). On the existence of solution of Diophantine equation 211ð›¼+229ð›½=ð›¾2, International Journal of Interdisciplinary Global Studies, 14(4), 78-79.
9. Rabago, J.F.T. (2013) On an open problem by B. Sroysang, Konuralp Journal of Mathematics, 1(2), 30-32.
10. Aggarwal, S. and Sharma, N. (2020). On the non-linear Diophantine equation 379ð‘¥+397ð‘¦=ð‘§2, Open Journal of Mathematical Sciences, 4(1), 397-399. DOI: 10.30538/oms2020.0129
11. Aggarwal, S. (2020). On the existence of solution of Diophantine equation 193ð‘¥+211ð‘¦=ð‘§2, Journal of Advanced Research in Applied Mathematics and Statistics, 5(3&4), 1-2.
12. Aggarwal, S. (2023). Solution of the Diophantine equation 323ð‘¥+85ð‘¦=ð‘§2, Journal of Advanced Research in Applied Mathematics and Statistics, 8(1 & 2), 6-9.
13. Aggarwal, S., Sharma, S.D. and Sharma, N. (2020). On the non-linear Diophantine equation 313ð‘¥+331ð‘¦=ð‘§2, Journal of Advanced Research in Applied Mathematics and Statistics, 5(3&4), 3-5.
14. Aggarwal, S., Sharma, S.D. and Chauhan, R. (2020). On the non-linear Diophantine equation 331ð‘¥+349ð‘¦=ð‘§2, Journal of Advanced Research in Applied Mathematics and Statistics, 5(3&4), 6-8.
15. Aggarwal, S. and Kumar, S. (2021). On the exponential Diophantine equation (132ð‘š)+(6ð‘Ÿ+1)ð‘›=ð‘§2, Journal of Scientific Research, 13(3), 845-849.
16. Aggarwal, S. and Kumar, S. (2021). On the exponential Diophantine equation 439ð‘+457ð‘ž=ð‘Ÿ2, Journal of Emerging Technologies and Innovative Research, 8(3), 2357-2361.
17. Aggarwal, S. and Kumar, S. (2021). On the exponential Diophantine equation (72ð‘š)+(6ð‘Ÿ+1+1)ð‘›=ðœ”2, International Journal of Research and Scientific Innovation, 8(4), 58-60.
18. Aggarwal, S. and Kumar, S. (2021). On the exponential Diophantine equation (22ð‘š+1−1)+(6ð‘Ÿ+1)ð‘›=ð‘§2, International Journal of Research and Innovation in Applied Science, 6(4), 49-51.
19. Aggarwal, S. and Kumar, S. (2021). On the exponential Diophantine equation ð‘€3ð‘+ð‘€5ð‘ž=ð‘Ÿ2, International Journal of Research and Innovation in Applied Science, 6(3), 126-127.
20. Aggarwal, S. and Kumar, S. (2021). On the exponential Diophantine equation (192ð‘š)+(6ð›¾+1+1)ð‘›=ðœŒ2, International Journal of Research and Innovation in Applied Science, 6(2), 112-114.
21. Aggarwal, S. and Kumar, S. (2021). On the exponential Diophantine equation (192ð‘š)+(12ð›¾+1)ð‘›=ðœŒ2, International Journal of Research and Innovation in Applied Science, 6(3), 14-16.
22. Aggarwal, S. and Kumar, S. (2021). On the exponential Diophantine equation (192ð‘š)+(6ð›¾+1)ð‘›=ðœŒ2, International Journal of Research and Innovation in Applied Science, 6(3), 128-130.
23. Aggarwal, S. and Kumar, S. (2021). On the non-linear Diophantine equation [19]2ð‘š+[22ð‘Ÿ+1−1]=ðœŒ2, International Journal of Latest Technology in Engineering, Management & Applied Science, 10 (2), 14-16.
24. Aggarwal, S. (2021). On the exponential Diophantine equation (22ð‘š+1−1)+(13)ð‘›=ð‘§2, Engineering and Applied Science Letter, 4(1), 77-79.
25. Aggarwal, S., Swarup, C., Gupta, D. and Kumar, S. (2022). Solution of the Diophantine equation 143ð‘¥+45ð‘¦=ð‘§2, Journal of Advanced Research in Applied Mathematics and Statistics, 7(3&4), 1-4.
26. Goel, P., Bhatnagar, K. and Aggarwal, S. (2020). On the exponential Diophantine equation ð‘€5ð‘+ð‘€7ð‘ž=ð‘Ÿ2, International Journal of Interdisciplinary Global Studies, 14(4), 170-171.
27. Kumar, S., Bhatnagar, K., Kumar, A. and Aggarwal, S. (2020). On the exponential Diophantine equation (22ð‘š+1−1)+(6ð‘Ÿ+1+1)ð‘›=ðœ”2, International Journal of Interdisciplinary Global Studies, 14(4), 183-184.
28. Bhatnagar, K. and Aggarwal, S. (2020). On the exponential Diophantine equation 421ð‘+439ð‘ž=ð‘Ÿ2, International Journal of Interdisciplinary Global Studies, 14(4), 128-129.
29. Kumar, S., Bhatnagar, K., Kumar, N. and Aggarwal, S. (2020). On the exponential Diophantine equation (72ð‘š)+(6ð‘Ÿ+1)ð‘›=ð‘§2, International Journal of Interdisciplinary Global Studies, 14(4), 181-182
30. Gupta, D., Kumar, S. and Aggarwal, S. (2022). Solution of non-linear exponential Diophantine equation ð‘¥ð›¼+(1+ð‘šð‘¦)ð›½=ð‘§2, Journal of Emerging Technologies and Innovative Research, 9(9), d486-d489.
31. Gupta, D., Kumar, S. and Aggarwal, S. (2022). Solution of non-linear exponential Diophantine equation (ð‘¥ð‘Ž+1)ð‘š+(ð‘¦ð‘+1)ð‘›=ð‘§2, Journal of Emerging Technologies and Innovative Research, 9(9), f154-f157.
32. Aggarwal, S. and Upadhyaya, L.M. (2022). On the Diophantine equation 8ð›¼+67ð›½=ð›¾2, Bulletin of Pure & Applied Sciences-Mathematics and Statistics, 41(2), 153-155.
33. Aggarwal, S. and Upadhyaya, L.M. (2023). Solution of the Diophantine equation 22ð‘¥+40ð‘¦=ð‘§2, Bulletin of Pure & Applied Sciences-Mathematics and Statistics, 42(2), 122-125.
34. Aggarwal, S. and Upadhyaya, L.M. (2023). Solution of the Diophantine equation 783ð‘¥+85ð‘¦=ð‘§2, Bulletin of Pure & Applied Sciences-Mathematics and Statistics, 42(1), 31-35.
35. Aggarwal, S., Upadhyaya, L.M. and Shahida, A.T. (2024). On the Diophantine equation 8ð’¾+71ð’¿=ð“€2, Journal of Advanced Research in Applied Mathematics and Statistics, 9(1 & 2), 1-4.
36. Aggarwal, S., Swarup, C., Gupta, D. and Kumar, S. (2023). Solution of the Diophantine equation 143ð‘¥+85ð‘¦=ð‘§2, International Journal of Progressive Research in Science and Engineering, 4(2), 5-7.
37. Aggarwal, S. and Shahida, A.T. (2023). Solution of the Diophantine equation 10ð‘¥+40ð‘¦=ð‘§2, Journal of Advanced Research in Applied Mathematics and Statistics, 8(3 & 4), 26-29.
38. Aggarwal, S. and Shahida, A.T. (2024). Solution of exponential Diophantine equation ð‘›ð‘¥+43ð‘¦=ð‘§2, where ð‘›â‰¡2(ð‘šð‘œð‘‘ 129) and ð‘›+1 is not a perfect square, Journal of Scientific Research, 16(2), 429-435.
39. Aggarwal, S., Kumar, S., Gupta, D. and Kumar, S. (2023). Solution of the Diophantine equation 143ð‘¥+485ð‘¦=ð‘§2, International Research Journal of Modernization in Engineering Technology and Science, 5(2), 555-558.
40. Aggarwal, S., Shahida, A. T., Pandey, E. and Vyas, A. (2023). On the problem of solution of non-linear (exponential) Diophantine equation ð›½ð‘¥+(ð›½+18)ð‘¦=ð‘§2, Mathematics and Statistics, 11(5), 834-839.
41. Aggarwal, S., Pandey, R. and Kumar, R. (2024). Solution of the exponential Diophantine equation 10ð‘¥+400ð‘¦=ð‘§2, International Journal of Latest Technology in Engineering, Management & Applied Science, 13 (2), 38-40.
42. Sroysang, B. (2014) On the Diophantine equation 8ð‘¥+13ð‘¦=ð‘§2, International Journal of Pure and Applied Mathematics, 90(1), 69-72.
43. Sroysang, B. (2012) On the Diophantine equation 31ð‘¥+32ð‘¦=ð‘§2, International Journal of Pure and Applied Mathematics, 81(4), 609-612.
2. Singh, K. (2020) Number theory step by step, Oxford University Press, United Kingdom.
3. Schoof, R. (2008) Catalan’s conjecture, Springer-Verlag, London.
4. Aggarwal, S., Sharma, S.D. and Singhal, H. (2020). On the Diophantine equation 223ð‘¥+241ð‘¦=ð‘§2, International Journal of Research and Innovation in Applied Science, 5 (8), 155-156.
5. Aggarwal, S., Sharma, S.D. and Vyas, A. (2020). On the existence of solution of Diophantine equation 181ð‘¥+199ð‘¦=ð‘§2, International Journal of Latest Technology in Engineering, Management & Applied Science, 9 (8), 85-86.
6. Kumar, A., Chaudhary, L. and Aggarwal, S. (2020). On the exponential Diophantine equation 601ð‘+619ð‘ž=ð‘Ÿ2, International Journal of Interdisciplinary Global Studies, 14(4), 29-30.
7. Sroysang, B. (2012) More on the Diophantine equation 8ð‘¥+19ð‘¦=ð‘§2, International Journal of Pure and Applied Mathematics, 81(4), 601-604.
8. Mishra, R., Aggarwal, S. and Kumar, A. (2020). On the existence of solution of Diophantine equation 211ð›¼+229ð›½=ð›¾2, International Journal of Interdisciplinary Global Studies, 14(4), 78-79.
9. Rabago, J.F.T. (2013) On an open problem by B. Sroysang, Konuralp Journal of Mathematics, 1(2), 30-32.
10. Aggarwal, S. and Sharma, N. (2020). On the non-linear Diophantine equation 379ð‘¥+397ð‘¦=ð‘§2, Open Journal of Mathematical Sciences, 4(1), 397-399. DOI: 10.30538/oms2020.0129
11. Aggarwal, S. (2020). On the existence of solution of Diophantine equation 193ð‘¥+211ð‘¦=ð‘§2, Journal of Advanced Research in Applied Mathematics and Statistics, 5(3&4), 1-2.
12. Aggarwal, S. (2023). Solution of the Diophantine equation 323ð‘¥+85ð‘¦=ð‘§2, Journal of Advanced Research in Applied Mathematics and Statistics, 8(1 & 2), 6-9.
13. Aggarwal, S., Sharma, S.D. and Sharma, N. (2020). On the non-linear Diophantine equation 313ð‘¥+331ð‘¦=ð‘§2, Journal of Advanced Research in Applied Mathematics and Statistics, 5(3&4), 3-5.
14. Aggarwal, S., Sharma, S.D. and Chauhan, R. (2020). On the non-linear Diophantine equation 331ð‘¥+349ð‘¦=ð‘§2, Journal of Advanced Research in Applied Mathematics and Statistics, 5(3&4), 6-8.
15. Aggarwal, S. and Kumar, S. (2021). On the exponential Diophantine equation (132ð‘š)+(6ð‘Ÿ+1)ð‘›=ð‘§2, Journal of Scientific Research, 13(3), 845-849.
16. Aggarwal, S. and Kumar, S. (2021). On the exponential Diophantine equation 439ð‘+457ð‘ž=ð‘Ÿ2, Journal of Emerging Technologies and Innovative Research, 8(3), 2357-2361.
17. Aggarwal, S. and Kumar, S. (2021). On the exponential Diophantine equation (72ð‘š)+(6ð‘Ÿ+1+1)ð‘›=ðœ”2, International Journal of Research and Scientific Innovation, 8(4), 58-60.
18. Aggarwal, S. and Kumar, S. (2021). On the exponential Diophantine equation (22ð‘š+1−1)+(6ð‘Ÿ+1)ð‘›=ð‘§2, International Journal of Research and Innovation in Applied Science, 6(4), 49-51.
19. Aggarwal, S. and Kumar, S. (2021). On the exponential Diophantine equation ð‘€3ð‘+ð‘€5ð‘ž=ð‘Ÿ2, International Journal of Research and Innovation in Applied Science, 6(3), 126-127.
20. Aggarwal, S. and Kumar, S. (2021). On the exponential Diophantine equation (192ð‘š)+(6ð›¾+1+1)ð‘›=ðœŒ2, International Journal of Research and Innovation in Applied Science, 6(2), 112-114.
21. Aggarwal, S. and Kumar, S. (2021). On the exponential Diophantine equation (192ð‘š)+(12ð›¾+1)ð‘›=ðœŒ2, International Journal of Research and Innovation in Applied Science, 6(3), 14-16.
22. Aggarwal, S. and Kumar, S. (2021). On the exponential Diophantine equation (192ð‘š)+(6ð›¾+1)ð‘›=ðœŒ2, International Journal of Research and Innovation in Applied Science, 6(3), 128-130.
23. Aggarwal, S. and Kumar, S. (2021). On the non-linear Diophantine equation [19]2ð‘š+[22ð‘Ÿ+1−1]=ðœŒ2, International Journal of Latest Technology in Engineering, Management & Applied Science, 10 (2), 14-16.
24. Aggarwal, S. (2021). On the exponential Diophantine equation (22ð‘š+1−1)+(13)ð‘›=ð‘§2, Engineering and Applied Science Letter, 4(1), 77-79.
25. Aggarwal, S., Swarup, C., Gupta, D. and Kumar, S. (2022). Solution of the Diophantine equation 143ð‘¥+45ð‘¦=ð‘§2, Journal of Advanced Research in Applied Mathematics and Statistics, 7(3&4), 1-4.
26. Goel, P., Bhatnagar, K. and Aggarwal, S. (2020). On the exponential Diophantine equation ð‘€5ð‘+ð‘€7ð‘ž=ð‘Ÿ2, International Journal of Interdisciplinary Global Studies, 14(4), 170-171.
27. Kumar, S., Bhatnagar, K., Kumar, A. and Aggarwal, S. (2020). On the exponential Diophantine equation (22ð‘š+1−1)+(6ð‘Ÿ+1+1)ð‘›=ðœ”2, International Journal of Interdisciplinary Global Studies, 14(4), 183-184.
28. Bhatnagar, K. and Aggarwal, S. (2020). On the exponential Diophantine equation 421ð‘+439ð‘ž=ð‘Ÿ2, International Journal of Interdisciplinary Global Studies, 14(4), 128-129.
29. Kumar, S., Bhatnagar, K., Kumar, N. and Aggarwal, S. (2020). On the exponential Diophantine equation (72ð‘š)+(6ð‘Ÿ+1)ð‘›=ð‘§2, International Journal of Interdisciplinary Global Studies, 14(4), 181-182
30. Gupta, D., Kumar, S. and Aggarwal, S. (2022). Solution of non-linear exponential Diophantine equation ð‘¥ð›¼+(1+ð‘šð‘¦)ð›½=ð‘§2, Journal of Emerging Technologies and Innovative Research, 9(9), d486-d489.
31. Gupta, D., Kumar, S. and Aggarwal, S. (2022). Solution of non-linear exponential Diophantine equation (ð‘¥ð‘Ž+1)ð‘š+(ð‘¦ð‘+1)ð‘›=ð‘§2, Journal of Emerging Technologies and Innovative Research, 9(9), f154-f157.
32. Aggarwal, S. and Upadhyaya, L.M. (2022). On the Diophantine equation 8ð›¼+67ð›½=ð›¾2, Bulletin of Pure & Applied Sciences-Mathematics and Statistics, 41(2), 153-155.
33. Aggarwal, S. and Upadhyaya, L.M. (2023). Solution of the Diophantine equation 22ð‘¥+40ð‘¦=ð‘§2, Bulletin of Pure & Applied Sciences-Mathematics and Statistics, 42(2), 122-125.
34. Aggarwal, S. and Upadhyaya, L.M. (2023). Solution of the Diophantine equation 783ð‘¥+85ð‘¦=ð‘§2, Bulletin of Pure & Applied Sciences-Mathematics and Statistics, 42(1), 31-35.
35. Aggarwal, S., Upadhyaya, L.M. and Shahida, A.T. (2024). On the Diophantine equation 8ð’¾+71ð’¿=ð“€2, Journal of Advanced Research in Applied Mathematics and Statistics, 9(1 & 2), 1-4.
36. Aggarwal, S., Swarup, C., Gupta, D. and Kumar, S. (2023). Solution of the Diophantine equation 143ð‘¥+85ð‘¦=ð‘§2, International Journal of Progressive Research in Science and Engineering, 4(2), 5-7.
37. Aggarwal, S. and Shahida, A.T. (2023). Solution of the Diophantine equation 10ð‘¥+40ð‘¦=ð‘§2, Journal of Advanced Research in Applied Mathematics and Statistics, 8(3 & 4), 26-29.
38. Aggarwal, S. and Shahida, A.T. (2024). Solution of exponential Diophantine equation ð‘›ð‘¥+43ð‘¦=ð‘§2, where ð‘›â‰¡2(ð‘šð‘œð‘‘ 129) and ð‘›+1 is not a perfect square, Journal of Scientific Research, 16(2), 429-435.
39. Aggarwal, S., Kumar, S., Gupta, D. and Kumar, S. (2023). Solution of the Diophantine equation 143ð‘¥+485ð‘¦=ð‘§2, International Research Journal of Modernization in Engineering Technology and Science, 5(2), 555-558.
40. Aggarwal, S., Shahida, A. T., Pandey, E. and Vyas, A. (2023). On the problem of solution of non-linear (exponential) Diophantine equation ð›½ð‘¥+(ð›½+18)ð‘¦=ð‘§2, Mathematics and Statistics, 11(5), 834-839.
41. Aggarwal, S., Pandey, R. and Kumar, R. (2024). Solution of the exponential Diophantine equation 10ð‘¥+400ð‘¦=ð‘§2, International Journal of Latest Technology in Engineering, Management & Applied Science, 13 (2), 38-40.
42. Sroysang, B. (2014) On the Diophantine equation 8ð‘¥+13ð‘¦=ð‘§2, International Journal of Pure and Applied Mathematics, 90(1), 69-72.
43. Sroysang, B. (2012) On the Diophantine equation 31ð‘¥+32ð‘¦=ð‘§2, International Journal of Pure and Applied Mathematics, 81(4), 609-612.
Published
2024-12-23
How to Cite
AGGARWAL, Sudhanshu; UPADHYAYA, Lalit Mohan; CHAUBEY, Vidya Sagar.
The Study of the Exponential Diophantine Equation (ðŸ‘ðŸ•)ðŸð’™+(ðŸ”ð’“+ðŸ)ð’š=ð’›ðŸ.
Journal of Advanced Research in Applied Mathematics and Statistics, [S.l.], v. 9, n. 3&4, p. 48-53, dec. 2024.
ISSN 2455-7021.
Available at: <http://thejournalshouse.com/index.php/Journal-Maths-Stats/article/view/1352>. Date accessed: 02 jan. 2026.
Section
Research Article
