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I am desparately searching for a case that would skip the following conjecture (a variation of the Goldbach conjecture):

"Let $N$ an even integer, $P$ the very next prime smaller than $N$, and $D=N-P$. Then $D$ is always a prime. (Except $D=1$)"

Can anybody help me with a case to reject this conjecture?

Thank you in advance.

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2 Answers 2

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$122$ is even, and it is between $113$ and $127$. The difference, $122-113$, is $9$, definitely composite.

How I searched: primes greater than two are odd, so the difference between an even number and a prime is odd, so what is the smallest composite odd number? Then, the search was for a pair of neighboring primes at least nine apart.

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The counterexample of Kyle Miller solve the problem, but we can say more. Since we can take prime gaps abitrarily large, we have that exists infinite even numbers such that $D$ is composite.

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    $\begingroup$ How does the existence of arbitrarily large prime gaps imply that the difference $D$ between an even number and its immediately preceding prime number is composite infinitely often? $\endgroup$
    – Jeppe Stig Nielsen
    Sep 16, 2016 at 10:17
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    $\begingroup$ @JeppeStigNielsen Beacuse you can choose an even number $N$ such that its distance with the next prime smaller than $N$ is an odd composite number. $\endgroup$
    – Marco Cantarini
    Sep 16, 2016 at 10:40
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    $\begingroup$ @JeppeStigNielsen Assume that the distance from $p_{n+1}$ and $p_{n}$ is greater than $10^{1000}$. Then consider $N=p_{n}+9$. It is an even number, the next smallest prime is $p_{n}$ and $D=9$. Now take $N=p_{n}+15$ and so on. And when you have considered all the possibilities, you can take a larger gaps between primes. $\endgroup$
    – Marco Cantarini
    Sep 16, 2016 at 11:52
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    $\begingroup$ Ah, of course, I do not know why I did not see that before! We see that all odd numbers (including odd composite numbers) occur infinitely often as $D$. $\endgroup$
    – Jeppe Stig Nielsen
    Sep 16, 2016 at 15:09
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    $\begingroup$ @MarcoCantarini - For the statement in the answer itself, you don't need the full strength of the arbitrarily large prime gaps result, just that prime gaps $>9$ occur infinitely often. But you seem to even be showing that every odd composite occurs as $D$ for some choice of $N$. $\endgroup$
    – Ben Blum-Smith
    Sep 17, 2016 at 6:36

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