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. LipiĆski) and “Partitions of primes by Chebyshev polynomials”

(arXiv:1806.09446).

The subject matter is quite elementary, no knowledge of Algebraic Number Theory

is expected.