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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."

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    I just now re-edited and re-tagged your question asking for reference requests too, so that proficient users can give you some good references @mahi2011-12-25

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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.

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    Is there any problem with my answer ? , it annoys the person who answers the question, if he doesn't hear any response even after taking strains to answer ( may be positive or negative response ) .2011-12-26
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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
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    @joriki I was talking about the probability that a number chosen according to some distribution is prime. Your last sentence seems to confirm that you agree that if we don't fix a distribution, we cannot talk about the probability of a number being prime, and that the experiment of asking people on the street has little to do with mathematics. I am not sure right now where the disagreement is.2011-12-26