| In 1980 the author had introduced a new function in number theory. In the London journal "Personal Computer World" it was later called Smarandache function. Many specialists on mathematics and information from different countries are intensively studying this function and adjacent problems. The most active are: H. Ibstedt (Sweden), P. Gronas (Norway), J. Duncan, J. C. MacCarthy, J. R. Sutton (England), K. Tauscher (Australia), T. Martin, Ch. Ashbacher (USA), M. Costewitz (France) and others.; In the thesis three new functions in the number theory are introduced: {dollar}eta, L, Psb{lcub}k{rcub}.{dollar} Formula to compute the values of these functions are determined. Several properties of these functions are established. With the help of the function {dollar}eta{dollar} the prime numbers are characterized. The function L is used to generalize the well known theorems in the number theory: Wilson, Fermat, Euler, Gauss, Lagrange, Leibniz, Moser, Sierpinsky. The functions {dollar}Psb{lcub}k{rcub}{dollar} answer the question: when several numbers are all prime. All three functions are involved in the solution of different problems of the number theory in connection with other number functions.; The obtained results can be used in the number theory and in adjacent domains, of modern algebra. |
