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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 $g_n=1$?)

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    $\begingroup$ All primes except $2$ are odd, so their difference is even. $\endgroup$ – studiosus Sep 25 '18 at 15:02
  • $\begingroup$ Pretty sure it is impossible since all multiples of 2 are all obviously even and not prime $\endgroup$ – csch2 Sep 25 '18 at 15:03
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    $\begingroup$ 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. $\endgroup$ – Tanner Swett Sep 25 '18 at 15:05
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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.

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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...

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