On some mathematical connections between Fermat’s Last Theorem, Modular Functions, Modular Elliptic Curves and some sector of String Theory

This paper is fundamentally a review, a thesis, of principal results obtained in some sectors of Number Theory and String Theory of various authoritative theoretical physicists and mathematicians. Precisely, we have described some mathematical results regarding the Fermat’s Last Theorem, the Mellin transform, the Riemann zeta function, the Ramanujan’s modular equations, how primes and adeles are related to the Riemann zeta functions and the p-adic and adelic string theory. Furthermore, we show that also the fundamental relationship concerning the Palumbo-Nardelli model (a general relationship that links bosonic string action and superstring action, i.e. bosonic and fermionic strings in all natural systems), can be related with some equations regarding the p-adic (adelic) string sector. Thence, in conclusion, we have described some new interesting connections that are been obtained between String Theory and Number Theory, with regard the arguments above mentioned. In the Chapters 1 and 2, we have described the mathematics concerning the Fermat’s Last Theorem, precisely the Wiles approach in the Chapter 1 and further mathematical aspects concerning the Fermat’s Last Theorem, precisely the modular forms, the Euler products, the Shimura map and the automorphic L-functions in the Chapter 2. Furthermore. In this chapter, we have described also some mathematical applications of the Mellin transform, only mentioned in the Chapter 1, the zeta-function quantum field theory and the quantum L-functions. In the Chapter 3, we have described how primes and adeles are related to the Riemann zeta function, precisely the Connes approach. In the Chapter 4, we have described the p-adic and adelic strings, precisely the open and closed p-adic strings, the adelic strings, the solitonic q-branes of padic string theory and the open and closed scalar zeta strings. In the Chapter 5, we have described some correlations obtained between some solutions in string theory, Riemann zeta function and Palumbo-Nardelli model. Precisely, we have showed the cosmological solutions from the D3/D7 system, the classification and stability of cosmological solutions, the solution applied to ten dimensional IIB supergravity, the connections with some equations concerning the Riemann zeta function, the Palumbo-Nardelli model and the Ramanujan’s identities. Furthermore, we have described the interactions between intersecting D-branes and the general action and equations of motion for a probe D3-brane moving through a type IIB supergravity background. Finally, in the Chapter 6, we have showed the connections between the equations of the various chapters.