Q:

Do we know of more than one occurance where Prime Gap=1?

A prime gap $g_n$ is the difference between two prime numbers, and as we know, the first two primes are 2 and 3, thus their prime gap is 1;
$$
g_n = p_{n+1}-p_n=\big\{ n=1 \big\}=3 -2=1.
$$
But have we found any other occurrences, except for the gap between primes 2 and 3, where the prime gap is 1? 
(For example the  twin prime conjecture shows that $g_n = 2$ for infinitely many integers $n$, but is there anything about $g_n=1$?)

A:

It is impossible for any two prime numbers $p,q\geq 3$ to have difference one. This would mean that a number $p$ is prime and $p+1$ is also prime. But one of them then would have to be even so cannot be a prime number, if you assume them to be at least three.

