Seminar at SMU Delhi

One of the important and difficult problems in mathematics is to find explicit integral solutions of Diophantine equations. There are several Diophantine equations such as Fermat's equation, Catalan's equation etc which shaped up the modern path of number theory in a great deal. In this talk we will consider a certain Diophantine equation called Sch\"{a}ffer's equation and its variations. This diophantine equation is well studied in the literature. Here we aim to find all the solutions of the following equation explicitly \begin{equation} (x+1)^3 - (x+2)^3 + \cdots +(-1)^{r-1} (x+r)^3 = y^n \end{equation} in variables $x,y,n$. with $r=2d+1$ is any positive odd integer satisfying $ 2\leq d \leq 50$ and $n\in\mathbb{N}$. This is a joint work with P.K Dey, S.S Rout and B. Maji.