1 On the Andrica and Cramer s Conjectures. Mathematical connections between Number Theory and some sectors of String Theory Rosario Turco, Maria Colonnese, Michele Nardelli 2 , 1 1 Dipartimento di Scienze della Terra Università degli Studi di Napoli Federico II, Largo S. Marcellino, 10 80138 Napoli, Italy 2 Dipartimento di Matematica ed Applicazioni R. Caccioppoli Università degli Studi di Napoli Federico II Polo delle Scienze e delle Tecnologie Monte S. Angelo, Via Cintia (Fuorigrotta), 80126 Napoli, Italy Abstract In this paper we have described, in the Section 1, some mathematics concerning the Andrica s conjecture. In the Section 2, we have described the Cramer Shank Conjecture. In the Section 3, we have described some equations concerning the possible proof of the Cramer s conjecture and the related differences between prime numbers, principally the Cramer s conjecture and Selberg s theorem. In the Section 4, we have described some equations concerning the p-adic strings and the zeta strings. In the Section 5, we have described some equations concerning the -deformation in toroidal compactification for N = 2 gauge theory. In conclusion, in the Section 6, we have described some possible mathematical connections between various sectors of string theory and number theory. 1. The Andrica s Conjecture [1] In this section we will show some mathematics related to the Andrica s conjecture: 1 1 n n p p using some our results on Legendre s conjecture ([2]). Andrica s conjecture Andrica s conjecture is so defined: Andrica s Conjecture is a conjecture of Numbers Theory, concerning the gaps between two successive prime numbers, formulated by romeno s mathematician Dorin Andrica in 1986. It affirms that, for every couple of consecutive numbers p n and p n+1 , we have: