An Ensemble of Multivariable Higher Degree Diophantine and Transcendental Equations

A vast and fascinating field of mathematics in number theory is the subject of Diophantine equations consisting of the study of polynomial equations usually involving two or more parameters such that only solutions in integers are concentrated. The mathematical study of Diophantine problems that Diophantus initiated is Diophantine analysis. Diophantine problems have fewer equations than unknown variables and involve finding integers that work correctly for all equations.

In studies of Diophantine equations of degrees higher than two, significant success was attained only in the 20th century. There has been interest in determining all integer solutions to multi variables and higher degree Diophantine equations among mathematicians. In this context, for simplicity and brevity, one may refer (Carmichael.,1959, Dickson.,1952, Mordell.,1969, Gopalan et.al., 2012a, Gopalan et.al., 2015b, Gopalan et.al., 2024c, Mahalakshmi, Shanthi.,2023a, Mahalakshmi, Shanthi.,2023b, Mahalakshmi, Shanthi.,2023c, Sathiyapriya et.al., 2024a, Sathiyapriya et.al., 2024b, Shanthi.,2023a, Shanthi.,2023b, Shanthi, Mahalakshmi.,2023a, Shanthi, Mahalakshmi.,2023b, Shanthi, Mahalakshmi.,2023c, Shanthi, Gopalan.,2024a, Shanthi, Gopalan.,2024b, Thiruniraiselvi, Gopalan., 2024a, Vidhyalakshmi et.al., 2022a) for some binary and ternary quadratic Diophantine equations.

Note that, the non-algebraic equations can be solved by transforming it into an equivalent polynomial equation. Some transcendental equation in more than one unknown can be solved by separation of the unknowns reducing them to polynomial equations (Thiruniraiselvi, Gopalan., 2024b; Vidhyalakshmi et.al., 2021b).

The focus in this book is on solving multivariable higher degree Diophantine equations along with transcendental equations. These types of equations are significant since they concentrate on obtaining solutions in integers which satisfy the considered algebraic and transcendental equations. These solutions play a vital role in different area of mathematics & science and help us in understanding the significance of number patterns. This book contains a reasonable collection of special multivariable higher degree Diophantine problems & transcendental equations with three and five unknowns. The procedure in obtaining varieties of solutions in integers for the polynomial and transcendental Diophantine equations considered in this book are illustrated in an elegant manner.