Using logical functions in combinatorics and elementary number theory
by
Vladimir Shmagin
Dept. of Mathematics, Univ. of Aerocosmical Instrument-making, St. Petersburg, Russia

In our view, the most impressive application of logical functions in elementary number theory is the following formula allowing to generate n-th prime number p_n:

pn= (n+1)2+1 å m=0  sg æ è n+1 - m å k=2  ({(k-1)!}2- k[{(k-1)!}2/k]) ö ø ,

where p₀=2, p₁=3, ...; square brackets mean an integer part; sg is a logical function defined by sg(a)=1 if a > 0 and sg (a)=0 if a <= 0. We give another analytical formula for p_n without factorial:

pn= (n+1)2+1 å m=0  sg æ è n-1- m å i=3  \chii ö ø ,     \chii = [\surdi] Õ j=2  {sg (i-j[i/j])},

where now p₂=2, p₃=3, ... , n > 3. In this joint work with Yuri Chebrakov, we adduce a set of other examples, in which, for description of some numerical sequences, combinatorial and number theoretic algorithms, analytical formulae are constructed by using logical functions.

Date received: April 13, 1999

Copyright © 1999 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cacf-28.