Integer Solutions of Bi quadratic Equation xy+(sigma^2+1)z^2=65w^4

In this work, we have shown that the non-homogeneous biquadratic equation has an unlimited number of non-zero integral solutions. Given the variety of biquadratic equations, the corresponding integer solutions for the other types of biquadratic equations with variables larger than or equal to 4 can be obtained.

[1] L.E.Dickson, History of Theory of Numbers, Chelsea Publishing company, New York, Vol.11, (1952). [2] L.J.Mordell, Diophantine equations, Academic Press, London(1969). [3] Andreescu, T.Andrica, D., An Intrduction to Diophantine Equations, GIL Publishing house, 2002. [4]  Janaki G and  Vidhya S. On the Integer Solutions of the Homogeneous Biquadratic Diophantine Equation . ISSN 2321 3361 © 2016 IJESC Research Article Volume 6 Issue No. 6. 7275-7278. [5]  Saranya, C., & Janaki, G. (2021). Integral Solutions of the Binary Quadratic Diophantine Equation. Recent Advances in Mathematical Research and Computer Science Vol. 2, 83-90. [6] Saranya P , Poorani K. Intrinsic solutions of Diophantine Equation Involving Centered Square Number. International Journal for Research in Applied Science & Engineering Technology (IJRASET) ISSN: 2321-9653; Volume 11 Issue III Mar 2023 [7] Janaki, G., & Saranya, C. (2020). Integral Solutions of the Non-Homogeneous Sextic Equation with three Unknowns. Int. J. Sci. Res. in Multidisciplinary Studies Vol, 6(3). [8] Shankari, A. G., Janaki, G., & Saranya, A. (2025). Integral Solutions of the Homogeneous Quinary Quintic Diophantine Equation Involving Palindrome Number p5+ q5− p2q2 (p+ q)= 939 (p+ q)(r− s) 2t2. Arya Bhatta Journal of Mathematics and Informatics, 17(1), 43-48. [9] Janaki G,  Sangeetha P, On the Ternary Quadratic Diophantine     Equation    , Arya  Bhatta  Journal  of  Mathematics  and Informatics, Vol (16), Issue 01, 49-54, 2024. [10] Saranya, P., & Janaki, G. (2017). On the Quintic Non-Homogeneous  Diophantine  Equation. International  Journal  of  Engineering  Science, Volume 11 (III), 4685