Periodic points of the Lotka-Volterra map and some open problems in theory of numbers

We consider a two dimensional dynamical system system given
by the map F(x,y) = ( x(4-x?y), xy ) which maps the triangle

{(x,y): 0<=x, 0<=y, x + y<=4}

onto itself.
The aim of our talk is to show that properties of periodic points
of the map F are closely related to some open problems in theory of numbers.
These open problems are Artin's conjecture on primitive roots, Wieferich primes
and Sophie Germain primes.