On some equations concerning the M-Theory and Topological strings and the Gopakumar- Vafa formula applied in some sectors of String Theory and Number Theory

In the present paper we have described in the Chapter 1 some equations concerning the M-Theory, the Topological strings and the Topological Gauge Theory, in the Chapter 2 some equations concerning the Gopakumar-Vafa formula in Type IIA compactification to four dimensions on a Calabi-Yau manifold in terms of a counting of BPS states in M-theory. Finally, in the Chapter 3, we have described some possible methods of factorization and their various possible mathematical connections concerning the solutions for some equations regarding the above sectors of string theory