Question 83
Prove that the positive integer $n$ is the product of exactly two primes that differ by 2 if and only if
$$
\phi(n)\sigma(n) = (n-3)(n+1).
$$
(Recall that $\phi(n)$ equals the number of positive integers less than or equal to $n$, relatively prime to $n$, and $\sigma(n)$ equals the sum of the positive divisors of $n$, including 1 and $n$.)
$$
\phi(n)\sigma(n) = (n-3)(n+1).
$$
(Recall that $\phi(n)$ equals the number of positive integers less than or equal to $n$, relatively prime to $n$, and $\sigma(n)$ equals the sum of the positive divisors of $n$, including 1 and $n$.)