A question based on prime numbers. Show that if $P$ and $8P - 1$ are prime, then $8P + 1$ is composite. First of all I analyzed that except for the case of 2 & 3, the minimum difference between two prime numbers is always greater than or equal to 2. Then via shrewd deduction I found that $P = 3$ satisfies the condition. But I could not find any means to generalise it. I'd appreciate some help.

**Hint:** All prime numbers except for $2$ and $3$ are of the form $6n\pm1$. In your case, if $8p\pm1$ are both prime, then $8p=6n\iff4p=3n$. Can you conclude ?