Question 32
a) Find all positive integers $g$ with the following property: for each odd prime number $p$ there exists a positive integer $n$ such that $p$ divides the two integers
$$
g^{n}-n \quad \text{ and } \quad g^{n+1}-(n+1)
$$
b) Find all positive integers $g$ with the following property: for each odd prime number $p$ there exists a positive integer $n$ such that $p$ divides the two integers
$$
g^{n}-n^{2} \quad \text{ and } \quad g^{n+1}-(n+1)^{2}
$$
$$
g^{n}-n \quad \text{ and } \quad g^{n+1}-(n+1)
$$
b) Find all positive integers $g$ with the following property: for each odd prime number $p$ there exists a positive integer $n$ such that $p$ divides the two integers
$$
g^{n}-n^{2} \quad \text{ and } \quad g^{n+1}-(n+1)^{2}
$$