2. A positive integer $n$ is said to be 'good' if $n^{2}-1$ can be written as the product of three distinct prime numbers. Find the sum of the five smallest 'good' integers.
(1 mark)
對於正整數 $n$, 若 $n^{2}-1$ 可寫成三個不同質數之積, 則 $n$ 稱為「好數」。求最小的五個「好數」之和。
2. 104
2. Note that $n^{2}-1=(n-1)(n+1)$. We thus want one of $n-1$ and $n+1$ to be a prime and the other to be a product of two primes. All of these primes have to be odd since $n-1$ and $n+1$ have the same parity and there exists only one even prime.
Call a positive integer a 'semiprime' if it is the product of two distinct odd primes. The first five 'semiprimes' are thus $15,21,33,35,39$. We need to look for small primes which differ from a 'semiprime' by 2 . We thus, get $(13,15),(15,17),(19,21),(21,23)$ and $(31,33)$ as the five smallest sets of $(n-1, n+1)$. Therefore, the five smallest 'good' integers are $14,16,20$, 22 and 32 . Summing them up, we get the answer 104 .