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 completiond of the Robinson non-archimedean field and established basic algebraic properties of d [6]. In 1985 H. Gonshor established further fundamental properties of d [7].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 completiond in transcendental number theory were considered. We dealing using set theory ZFC (-model of ZFC).Given an class of analytic functions of one complex variable f z, we investigate the arithmetic nature of the values of fz at transcendental points en, n .Main results are: (i) the both numbers e and e are irrational, (ii) number ee is transcendental. Nontrivial generalization of the Lindemann- Weierstrass theorem is obtained.