MathDB

For all real number

x

consider the family

F(x)

of all sequences

(a_{n})_{n\geq 0}

satisfying the equation a_{n+1}=x-\frac{1}{a_{n}} (n\geq 0). A positive integer

p

is called a minimal period of the family

F(x)

if (a) each sequence

\left(a_{n}\right)\in F(x)

is periodic with the period

p

, (b) for each

0<q<p

there exists

\left(a_{n}\right)\in F(x)

such that

q

is not a period of

\left(a_{n}\right)

. Prove or disprove that for each positive integer

P

there exists a real number

x=x(P)

such that the family

F(x)

has the minimal period

p>P

Problem Statement