If we look at the difference between consecutive prime numbers, $p \gt 2$, it always appears to be an even number.
For example, here are the seven consecutive primes starting at the $10^{10th}$ prime.
$p_i = \{252097800623, 252097800629, 252097800637, 252097800667, 252097800737, 252097800743, 252097800839\}$
The differences between the consecutive primes above are $\{6, 8, 30, 70, 6, 96\}$, and are all an even number .
This, of course, is automatic for twin primes since by definition they differ by $2$.
Also, this holds for all balanced primes, A006562 – Balanced primes, since we have $2*p_n = p_{n-1} + p_{n+1}$.
There is a table of such values in A001223 – Differences between consecutive primes on OEIS.
My questions are:
(1) Is it considered a conjecture that the difference between consecutive primes $p \gt 2$ is always an even number?
I wasn't sure if there was some argument regarding Prime Gaps that guarantees such a result and it is easy.
(2) Has this been proven?
Note that I found the Prime Difference Function, but is that the latest?
Regards
Best Answer
You are thinking way too hard about this.
First, to the question in the title (but not as asked in the text) no: $3-2=1$.
As asked in the text for odd primes, then the difference between two odd numbers is always an even number: $$2p+1 - (2q+1) = 2(p-q).$$