A friend of mine gave me a number theoretical problem the other day:
Show that there are infinitely many solutions to this equation ($n\in \mathbb{N}$, $\phi$ is Euler's totient function):
$\phi(n+2)=\phi(n)+2$.
What I know is that twin primes $(n,n+2)$, doubles of Mersenne primes $n=2\cdot (2^p - 1)$ where $2^p-1$ is a Mersenne prime, or Sophie Germain primes $n=4p$ where $p$ and $2p+1$ are both primes, satisfy this equation. But as far as I know, we don't know if there are infinitely many of any of these. (Correct me if I'm wrong...)
Can anyone give me a clue how to prove/solve this exercise?
The exercise demands a (rigorous) proof and there is none , unless I missed infinite many "sporadic" solutions , solutions not fitting in one of the three families (although upto $10^9$ , the only one is $18$).
Although , almost surely there are infinite many twin-primes and also extremely likely infinite many Sophie-Germain-primes and finally probably infinite many Mersenne-primes , we cannot rule out that all families are finite , so I agree you and see no possibility for an unconditional proof.