Is the difference between consecutive prime numbers always an even number? 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
 A: 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:
$$2p+1 - (2q+1) = 2(p-q).$$
A: Think binary. For all odd primes the last digit, in binary, is 1. So, for any two odd primes:
P1 = ...1
P2 = ...1
X = P1 - P2 = (+/-)...0
Where ... represents whatever leading digits there are.
A: Yes, always. If we get two prime numbers $\gt 2$, then both are odd and difference two odd numbers is always even. This can be simply proven: 
Each prime number $\gt 2$ is odd. This can be proven by contradiction: If those number is even, then this number is divisible by $2$ and then it is not a prime number.

Difference two odd numbers is even always. Let us have some two odd numbers  $p_1$ and $p_2$. These numbers are odd and according to this we can express these numbers as $p_1=2k+1$ and $p_2=2l+1$, where $k\ge 1$ and $l \ge k$, then $\Delta = p_2 - p_1 = 2l+1-2k-1=2(l-k)$. This number is even and difference two odd, including prime, numbers($\gt 2$) is always even number.
A: Yes this is absolutely  true ,Difference between two consecutive or any  prime numbers will always an even number. iff number is not 2 & 3.
Proof argument logic, There is common fact that the difference between any two odd numbers must be an even number & any prime number must be odd number so  difference between any prime number other than 2,3 will be even number.
Thanks
Anil Kesarwani
