Would consecutive prime integers be 2 number's apart like 1,2,3 and 5?

Question

I know it's kinda stupid but I would like to learn about integers, asking here cause I couldn't find it online. Just want an explanation, please.

$1$ is not a prime, your question is the Twin prime conjecture which is still open problem, the thing we know for sure is that infinitely many primes occur with gap $\leq 246$. — Ahmad, Nov 20 '20 at 20:40

score 0 · Answer 1 · answered Nov 20 '20 at 20:53

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Interpreting your question as : why are some of the first primes two numbers apart.

Because $2$ is prime so after $2$ no multiple of $2$ can be prime, so a few of the next primes after $2$ are going to be consecutive odd numbers.

answered Nov 20 '20 at 20:53

Quarantiner

128

score 0 · Accepted Answer · answered Nov 20 '20 at 21:10

First: $1$ is generally not considered to be prime. To be fair to you, it hasn't always been this way- many mathematicians considered $1$ prime until about 1900.

Second: No, not all consecutive prime numbers are two numbers apart. The first example is $2,3$, which are only one number apart. Another example is $7,11$, in which the number right in the middle of them, $9$, is not prime- $9=3^2$.

Third: However, as Ahmad mentioned in the comments, whether there are infinitely many primes that are two numbers apart is actually not known! It's called the twin prime conjecture, and it's one of the most famous unsolved problems in math today.

Would consecutive prime integers be 2 number's apart like 1,2,3 and 5?

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