Is the difference between consecutive prime numbers always an even number?

Question

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

Answer 1

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: $$2p+1 - (2q+1) = 2(p-q).$$