ON THE DIOPHANTINE EQUATION ππ²+πππ³=π΄π
Abstract
In this study, authors looked for non-negative integer solutions to the Diophantine equation 8πΎ+71πΏ=π2, where πΎ,πΏ,π are non-negative integers. For this, authors turned to Catalan's conjecture. The current paper's results demonstrate that there is only one non-negative integer solution to the Diophantine equation 8πΎ+71πΏ=π2, where πΎ,πΏ,π are non-negative integers. This solution is provided by (πΎ,πΏ,π )=(1,0,3).
AMS SUBJECT CLASSIFICATION: 11D61
References
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6. Sroysang, B. (2014) On the Diophantine equation 8π₯+13π¦=π§2, International Journal of Pure and Applied Mathematics, 90(1), 69-72.
7. Aggarwal, S. (2020) On the existence of solution of Diophantine equation 193π₯+211π¦=π§2, Journal of Advanced Research in Applied Mathematics and Statistics, 5(3&4), 1-2.
8. Sroysang, B. (2012) On the Diophantine equation 31π₯+32π¦=π§2, International Journal of Pure and Applied Mathematics, 81(4), 609-612.
9. Aggarwal, S. and Sharma, N. (2020) On the non-linear Diophantine equation 379π₯+397π¦=π§2, Open Journal of Mathematical Sciences, 4(1), 397-399. DOI: 10.30538/oms2020.0129
10. Bhatnagar, K. and Aggarwal, S. (2020) On the exponential Diophantine equation 421π+439π=π2, International Journal of Interdisciplinary Global Studies, 14(4), 128-129.
11. Gupta, D., Kumar, S. and Aggarwal, S. (2022) Solution of non-linear exponential Diophantine equation (π₯π+1)π+(π¦π+1)π=π§2, Journal of Emerging Technologies and Innovative Research, 9(9), f154-f157.
12. Gupta, D., Kumar, S. and Aggarwal, S. (2022) Solution of non-linear exponential Diophantine equation π₯πΌ+(1+ππ¦)π½=π§2, Journal of Emerging Technologies and Innovative Research, 9(9), d486-d489.
13. Sroysang, B. (2014) On the Diophantine equation 323π₯+325π¦=π§2, International Journal of Pure and Applied Mathematics, 91(3), 395-398.
14. Sroysang, B. (2014) On the Diophantine equation 3π₯+45π¦=π§2, International Journal of Pure and Applied Mathematics, 91(2), 269-272.
15. Sroysang, B. (2014) On the Diophantine equation 143π₯+145π¦=π§2, International Journal of Pure and Applied Mathematics, 91(2), 265-268.
16. Sroysang, B. (2014) On the Diophantine equation 3π₯+85π¦=π§2, International Journal of Pure and Applied Mathematics, 91(1), 131-134.
17. Sroysang, B. (2014) More on the Diophantine equation 4π₯+10π¦=π§2, International Journal of Pure and Applied Mathematics, 91(1), 135-138.
18. Aggarwal, S., Kumar, S., Gupta, D. and Kumar, S. (2023) Solution of the Diophantine equation 143π₯+485π¦=π§2, International Research Journal of Modernization in Engineering Technology and Science, 5(2), 555-558.
19. Kumar, A., Chaudhary, L. and Aggarwal, S. (2020) On the exponential Diophantine equation 601π+619π=π2, International Journal of Interdisciplinary Global Studies, 14(4), 29-30.
20. Mishra, R., Aggarwal, S. and Kumar, A. (2020) On the existence of solution of Diophantine equation 211πΌ+229π½=πΎ2, International Journal of Interdisciplinary Global Studies, 14(4), 78-79.
21. Aggarwal, S., Swarup, C., Gupta, D. and Kumar, S. (2023) Solution of the Diophantine equation 143π₯+85π¦=π§2, International Journal of Progressive Research in Science and Engineering, 4(2), 5-7.
22. Aggarwal, S., Swarup, C., Gupta, D. and Kumar, S. (2022) Solution of the Diophantine equation 143π₯+45π¦=π§2, Journal of Advanced Research in Applied Mathematics and Statistics, 7(3 & 4), 1-4.
23. Kumar, S., Bhatnagar, K., Kumar, A. and Aggarwal, S. (2020) On the exponential Diophantine equation (22π+1β1)+(6π+1+1)π=π2, International Journal of Interdisciplinary Global Studies, 14(4), 183-184.
24. Kumar, S., Bhatnagar, K., Kumar, N. and Aggarwal, S. (2020) On the exponential Diophantine equation (72π)+(6π+1)π=π§2, International Journal of Interdisciplinary Global Studies, 14(4), 181-182.
25. Aggarwal, S., Sharma, S.D. and Singhal, H. (2020) On the Diophantine equation 223π₯+241π¦=π§2, International Journal of Research and Innovation in Applied Science, 5 (8), 155-156.
26. Aggarwal, S., Sharma, S.D. and Vyas, A. (2020) On the existence of solution of Diophantine equation 181π₯+199π¦=π§2, International Journal of Latest Technology in Engineering, Management & Applied Science, 9 (8), 85-86.
27. Aggarwal, S., Shahida, A. T., Pandey, E. and Vyas, A. (2023) On the problem of solution of non-linear (exponential) Diophantine equation π½π₯+(π½+18)π¦=π§2, Mathematics and Statistics, 11(5), 834-839.
28. Schoof, R. (2008) Catalanβs conjecture, Springer-Verlag, London.
2. Andreescu, T. and Andrica, D. (2002) An introduction to Diophantine equations, GIL Publishing House, ISBN 973-9238-88-2.
3. Mordell, L.J. (1969) Diophantine equations, Academic Press, London, New York.
4. Sierpinski, W. (1988) Elementary theory of numbers, 2nd edition, North-Holland, Amsterdam.
5. Sroysang, B. (2012) More on the Diophantine equation 8π₯+19π¦=π§2, International Journal of Pure and Applied Mathematics, 81(4), 601-604.
6. Sroysang, B. (2014) On the Diophantine equation 8π₯+13π¦=π§2, International Journal of Pure and Applied Mathematics, 90(1), 69-72.
7. Aggarwal, S. (2020) On the existence of solution of Diophantine equation 193π₯+211π¦=π§2, Journal of Advanced Research in Applied Mathematics and Statistics, 5(3&4), 1-2.
8. Sroysang, B. (2012) On the Diophantine equation 31π₯+32π¦=π§2, International Journal of Pure and Applied Mathematics, 81(4), 609-612.
9. Aggarwal, S. and Sharma, N. (2020) On the non-linear Diophantine equation 379π₯+397π¦=π§2, Open Journal of Mathematical Sciences, 4(1), 397-399. DOI: 10.30538/oms2020.0129
10. Bhatnagar, K. and Aggarwal, S. (2020) On the exponential Diophantine equation 421π+439π=π2, International Journal of Interdisciplinary Global Studies, 14(4), 128-129.
11. Gupta, D., Kumar, S. and Aggarwal, S. (2022) Solution of non-linear exponential Diophantine equation (π₯π+1)π+(π¦π+1)π=π§2, Journal of Emerging Technologies and Innovative Research, 9(9), f154-f157.
12. Gupta, D., Kumar, S. and Aggarwal, S. (2022) Solution of non-linear exponential Diophantine equation π₯πΌ+(1+ππ¦)π½=π§2, Journal of Emerging Technologies and Innovative Research, 9(9), d486-d489.
13. Sroysang, B. (2014) On the Diophantine equation 323π₯+325π¦=π§2, International Journal of Pure and Applied Mathematics, 91(3), 395-398.
14. Sroysang, B. (2014) On the Diophantine equation 3π₯+45π¦=π§2, International Journal of Pure and Applied Mathematics, 91(2), 269-272.
15. Sroysang, B. (2014) On the Diophantine equation 143π₯+145π¦=π§2, International Journal of Pure and Applied Mathematics, 91(2), 265-268.
16. Sroysang, B. (2014) On the Diophantine equation 3π₯+85π¦=π§2, International Journal of Pure and Applied Mathematics, 91(1), 131-134.
17. Sroysang, B. (2014) More on the Diophantine equation 4π₯+10π¦=π§2, International Journal of Pure and Applied Mathematics, 91(1), 135-138.
18. Aggarwal, S., Kumar, S., Gupta, D. and Kumar, S. (2023) Solution of the Diophantine equation 143π₯+485π¦=π§2, International Research Journal of Modernization in Engineering Technology and Science, 5(2), 555-558.
19. Kumar, A., Chaudhary, L. and Aggarwal, S. (2020) On the exponential Diophantine equation 601π+619π=π2, International Journal of Interdisciplinary Global Studies, 14(4), 29-30.
20. Mishra, R., Aggarwal, S. and Kumar, A. (2020) On the existence of solution of Diophantine equation 211πΌ+229π½=πΎ2, International Journal of Interdisciplinary Global Studies, 14(4), 78-79.
21. Aggarwal, S., Swarup, C., Gupta, D. and Kumar, S. (2023) Solution of the Diophantine equation 143π₯+85π¦=π§2, International Journal of Progressive Research in Science and Engineering, 4(2), 5-7.
22. Aggarwal, S., Swarup, C., Gupta, D. and Kumar, S. (2022) Solution of the Diophantine equation 143π₯+45π¦=π§2, Journal of Advanced Research in Applied Mathematics and Statistics, 7(3 & 4), 1-4.
23. Kumar, S., Bhatnagar, K., Kumar, A. and Aggarwal, S. (2020) On the exponential Diophantine equation (22π+1β1)+(6π+1+1)π=π2, International Journal of Interdisciplinary Global Studies, 14(4), 183-184.
24. Kumar, S., Bhatnagar, K., Kumar, N. and Aggarwal, S. (2020) On the exponential Diophantine equation (72π)+(6π+1)π=π§2, International Journal of Interdisciplinary Global Studies, 14(4), 181-182.
25. Aggarwal, S., Sharma, S.D. and Singhal, H. (2020) On the Diophantine equation 223π₯+241π¦=π§2, International Journal of Research and Innovation in Applied Science, 5 (8), 155-156.
26. Aggarwal, S., Sharma, S.D. and Vyas, A. (2020) On the existence of solution of Diophantine equation 181π₯+199π¦=π§2, International Journal of Latest Technology in Engineering, Management & Applied Science, 9 (8), 85-86.
27. Aggarwal, S., Shahida, A. T., Pandey, E. and Vyas, A. (2023) On the problem of solution of non-linear (exponential) Diophantine equation π½π₯+(π½+18)π¦=π§2, Mathematics and Statistics, 11(5), 834-839.
28. Schoof, R. (2008) Catalanβs conjecture, Springer-Verlag, London.
Published
2024-02-03
How to Cite
AGGARWAL, Sudhanshu; UPADHYAYA, Lalit Mohan; A. T., Shahida.
ON THE DIOPHANTINE EQUATION ππ²+πππ³=π΄π.
Journal of Advanced Research in Applied Mathematics and Statistics, [S.l.], v. 9, n. 1&2, p. 1-4, feb. 2024.
ISSN 2455-7021.
Available at: <http://www.thejournalshouse.com/index.php/Journal-Maths-Stats/article/view/933>. Date accessed: 06 may 2024.
Section
Research Article
