# 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 $$g_n=1$$?)

• All primes except $2$ are odd, so their difference is even. – studiosus Sep 25 '18 at 15:02
• Pretty sure it is impossible since all multiples of 2 are all obviously even and not prime – csch2 Sep 25 '18 at 15:03
• Everyone, please keep in mind that if the answer to this question seems kind of obvious, that doesn't mean that it's a bad question. – Tanner Swett Sep 25 '18 at 15:05

## 2 Answers

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.

If the gap between two numbers is $$1$$, one of them is even. So if they are both prime numbers, one of them has to be $$2$$ (the only even prime number). It should answer your question...