Let $f ( x )$ be a polynomial with integer coefficients such that for each nonnegative integer $n , f ( n ) = \mathrm { a }$ perfect power of a prime number, i.e., of the form $p ^ { k }$, where $p$ is prime and $k$ a positive integer. ($p$ and $k$ can vary with $n$.) Show that $f$ must be a constant polynomial using the following steps or otherwise. a) If such a polynomial $f ( x )$ exists, then there is a polynomial $g ( x )$ with integer coefficients such that for each nonnegative integer $n , g ( n ) =$ a perfect power of a fixed prime number. b) Show that a polynomial $g ( x )$ as in part a must be constant.
Let $f ( x )$ be a polynomial with integer coefficients such that for each nonnegative integer $n , f ( n ) = \mathrm { a }$ perfect power of a prime number, i.e., of the form $p ^ { k }$, where $p$ is prime and $k$ a positive integer. ($p$ and $k$ can vary with $n$.) Show that $f$ must be a constant polynomial using the following steps or otherwise.\\
a) If such a polynomial $f ( x )$ exists, then there is a polynomial $g ( x )$ with integer coefficients such that for each nonnegative integer $n , g ( n ) =$ a perfect power of a fixed prime number.\\
b) Show that a polynomial $g ( x )$ as in part a must be constant.