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If p is a prime number, and k is an even integer, what is the probability p+k is a prime number?

According to my simulations p+108 is prime twice as often as p+344

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    Also see Dickson's conjecture. (Note typo above: first name should be Bateman.)2011-09-30

2 Answers 2

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Note that if $p \equiv 1 \mod 3$, $p + 344$ is divisible by 3 and so must be composite. On the other hand, $p + 108 \equiv p \mod 3$. Thus primes $p$ with $p+344$ prime can occur in only one residue class mod 3, but those with $p+108$ prime can occur in two residue classes mod 3. On the other hand, $p + 344 \equiv p \mod 43$. So I would expect that primes $p + 108$ would occur $(2/1) (41/42) = 41/21$ times as often as $p + 344$.

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    I did not say I had a proof. I was careful to use the wording "I would expect...".2011-10-02
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Technically, the answer to your question is zero. If you fix $k$ and take the number of primes $p$ less than $N$ such that $p+k$ is prime and divide it by the number of primes less than $N$ and take the limit as $N$ goes to infinity, you get zero.

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    Of course. Nevertheless, I'd interpret the question as being about asymptotics.2011-10-02