Non-Archimedean Analysis on the Extended Hyperreal Line *R_d and the Solution of Some Very Old Transcendence Conjectures over the Field Q

In 1980 F. Wattenberg constructed the Dedekind completion *R_d of the Robinson non-archimedean field *R and established basic algebraic properties of *R_d. In 1985 H. Gonshor established further fundamental properties of *R_d. In [4] important construction of summation of countable sequence of Wattenberg numbers was proposed and corresponding basic properties of such summation were considered. In this paper the important applications of the Dedekind completion *R_d in transcendental number theory were considered. Given any analytic function of one complex variable , we investigate the arithmetic nature of the values of at transcendental points . Main results are: 1) the both numbers and are irrational; 2) number e^e is transcendental. Nontrivial generalization of the Lindemann-Weierstrass theorem is obtained.