11

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.

al-Hwarizmi
  • 4,370
  • 2
  • 21
  • 36

2 Answers2

27

$122$ 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.

Kyle Miller
  • 20,247
16

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.

Community
  • 1
Marco Cantarini
  • 33,458
  • 1
    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? – Jeppe Stig Nielsen Sep 16 '16 at 10:17
  • 1
    @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. – Marco Cantarini Sep 16 '16 at 10:40
  • What I know, is that I can choose an even number $N$ such that its distance to the immediately preceding prime is as huge as I want. For example I can choose that $N$ such that the difference to the preceding prime is greater than $10^{1000}$. How can I use that to assure the difference is non-prime? – Jeppe Stig Nielsen Sep 16 '16 at 11:34
  • 7
    @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. – Marco Cantarini Sep 16 '16 at 11:52
  • 2
    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 $D$. – Jeppe Stig Nielsen Sep 16 '16 at 15:09
  • 1
    @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 $N$. – Ben Blum-Smith Sep 17 '16 at 06:36
  • @BenBlum-Smith Yes, you are right. – Marco Cantarini Sep 17 '16 at 06:52