A new way to study the linear recursive sequences of
order two will be presented. One of the highlights is the
discovery that the Fibonacci sequence
\[
x_{n+1} = x_n+x_{n-1}, x_0 =0, x_1 =1,
\]
has a “twin”
\[
y_{n+1} = 5y_n-5y_{n-1}, y_0 =0, y_1 =1.
\]
\
Their connection is that the even numbered elements of the two
sequences coincide (after the powers of 5 are factored out).
The odd numbered elements have disjoint sets of prime
divisors, both of prime density 1/3.
We will give applications to arithmetic properties of orbits of simple
dynamical systems: some rotations and a chaotic circle map.
The talk is based on the papers “On sequence groups” (arXiv:2110.00450)
(joint with Z. Lipiński) and “Partitions of primes by Chebyshev polynomials”
(arXiv:1806.09446).
The subject matter is quite elementary, no knowledge of Algebraic Number Theory
is expected.