Let
and
be two polynomials with integer coefficients, such that no nonconstant polynomial with rational coefficients divides both
and
Suppose that for every positive integer
the integers
and
are positive, and
divides
Prove that
is a constant polynomial.
Proposed by Oleksiy Klurman, Ukraine
%V0
Let $P(x)$ and $Q(x)$ be two polynomials with integer coefficients, such that no nonconstant polynomial with rational coefficients divides both $P(x)$ and $Q(x).$ Suppose that for every positive integer $n$ the integers $P(n)$ and $Q(n)$ are positive, and $2^{Q(n)}-1$ divides $3^{P(n)}-1.$ Prove that $Q(x)$ is a constant polynomial.
Proposed by Oleksiy Klurman, Ukraine
Školjka