Discussion on twin prime conjecture I understand where I am wrong in my previous post. Also, I am very thankful to all members, who answered and showed my errors in post. Now, I would like to know the proof for the following. 
"The difference of any two prime numbers $p$ and $q$ is expressible as $2k$; and the probability of finding primes $p$ at $n + k$ and $q$ at $n - k$ decreases as $n$ increases."
 A: You are referring to the generalized proof of " Twin-Prime Conjecture ", by using the Prime Number Theorem ( PNT ).
Prime number theorem states that  the density of primes is ruled by the law $$\pi(n) \sim \large\frac{n}{log(n)}$$ We can state this in a more precise form using Riemann's Li function, as $$\pi(n)=Li(n)+O(ne^{-a\sqrt{Log(n)}})$$ for some constant $a$.
we can compute that the probability of finding twin primes ( $a-b=2$ , but you are looking at $2k$ which will be an implication ) $a$ at $n + 1$ and $b$ at $n -1$ is about $\large\frac{2}{\rm{Log(n)}}$ , but you are writing about the case for some $k$ but here in this case I mentioned above its $k=1$ and the number of twin primes in the interval $n$ is about $\large\frac{2n}{\rm{Log(n)}^2}$.
So its clearly evident to say from the above things that 


*

*The difference of two prime numbers $a$ and $b$ is an even number $2k$.

*The probability of finding primes $a$ at $n + k$ and $b$ at $n - k$ decreases as $n$ increases.


The prime number theorem states that the number of primes less
than $n$ is asymptotic to $1/\rm{Log(n)}$. So if we choose a random integer
$m$ from the interval $[1,n]$, then the probability that $m$ is prime is
asymptotic to $1/\rm{Log(n)}$
So its crystal clear that one can apply the same thing to your conditions by adjusting the values and as the $n$ is in the denominator of probability its evident that they are inversely proportional.
I end here, and I seriously advice you to completely go through this and this.
But I find there are some more beautiful articles about this, but it takes some time for me to fish them out. I surely re-edit once if I find any of such things.
Thank you, 
Yours truly,
Iyengar.
A: There are several problems here


*

*The difference between an odd prime $p$ and 2 is not an even number

*If $n$ and $k$ have the same parity and $n>k+2$ then neither $n+k$ nor $n-k$ are prime

*You cannot talk about the probability of finding primes in this sense: a given number is either prime or not.  You might turn it into a probabilistic statement if you had a meaningful way of choosing $n$ and $k$ with some kind of distribution.  Or you could take the route of the Bateman-Horn conjecture
