# Is every composite number the average of two primes?

I'm interested in this question because it relates to a bad joke about people in their prime.

It seems to work for the first 20 numbers:

4 is the average of 5 and 3.
6 is the average of 5 and 7.
8 is the average of 5 and 11.
$\vdots$
16 is the average of 13 and 19.
$\vdots$

It would not be hard to write a program that checks more cases, but I suspect the hypothesis is true. If that is the case, then a more mathematical approach is needed.

• Nobody knows, this precisely Goldbach's conjecture! – Gamma Function Mar 10 '15 at 11:34
• Can we hear the joke please? – J.R. Mar 10 '15 at 11:40
• The joke: It's someones birthday and let's they're 20. Then you say "20-3 is prime and 20+3 is prime, so you're in the middle of the prime of your life!" – William Kurdahl Mar 10 '15 at 11:45
• @WilliamKurdahl To me, that could imply their life is from 20-3 to 20+3, but I'm probably reading it wrong (I get the joke though) – Cole Johnson Mar 10 '15 at 14:32
• It's been tested enough that the joke holds up for any currently living person. – aslum Mar 10 '15 at 14:48

So you are asking whether for every composite number $n$ there exist primes $p,q$ such that $$n=\frac{p+q}{2}$$ That is, $2n=p+q$, so you are asking whether $2n$ can be written as the sum of two primes.

The question whether every even integer greater than $2$ is a sum of two primes is a famous open problem known as the Goldbach conjecture.

• Of yes, I see that this is the Goldbach conjecture (poorly) disguised. Thanks to all who answered. – William Kurdahl Mar 10 '15 at 11:38
• Technically, OP's question is slightly weaker than Goldbach, because they assume $n$ is composite, while Goldbach does not. Does that assumption change anything? – Kevin Mar 10 '15 at 13:43
• @ColeJohnson Goldbach's asserts that any even number $2n$ can be written as the sum of two primes. But since OP restricts $n$ to be composite, OP's conjecture is less strong than Goldbach's conjecture, for e.g. Goldbach's states that $2 \cdot 7 = 14$ can be written as the sum of two primes, whereas OP's conjecture says nothing about $n = 7$ since $7$ is not composite. – Thomas Mar 10 '15 at 14:51
• On the other hand, it is rather trivial to show that every prime number is the average of two primes. I would consider the stated question equivalent to the Goldbach conjecture. – Umberto P. Mar 10 '15 at 16:02
• @Mehrad: Goldbach asks whether 2n can be written as the sum of two primes. We split into two cases: n is prime and n is composite. By solving the n is prime case (for instance by noting that 2n = n + n) we have reduced the Goldbach conjecture to the remaining case which is equivalent to the OP's question. Am I missing something? – Vincent Mar 11 '15 at 10:50

This is almost the same as the Goldbach conjecture, that every even number from four is the sum of two primes.
It has been checked into the quintillions (no, really!) but not proven. In 2013, a similar theorem was proven by Harald Helfgott, that every odd number from seven up is the sum of three primes.

• It was Vinogradov who first proved the weak Goldbach conjecture, at least for all large enough integers. – Anonymous999 Mar 11 '15 at 10:46
• @Anonymous999: "at least for" is misleading, at least it sounds that way to me; "only for" is more accurate. And "large enough" had only explicit bounds that were way, way too huge to have hope of reaching computationally, and thus more theoretical work was required to prove the theorem, provided by Helfgott. Certainly you're correct that Helfgott's result relied on Vinogradov. [I'm just relying on what experts have said about this.] – Jonas Meyer Mar 11 '15 at 12:44

A generalized equation would be in the form of:

$$x =\frac{P_1+P_2}{2}$$

Where $x \in Z^+$, and $P_1,P_2 \in Z^p$ (Here, I define $Z^p$ to be a prime number).

It can be rewritten as $$2x = P_1+P_2$$

Let $2x=a$ Then the equation is: $$a = P_1+P_2$$

I.e. Your question's generalised form ($x =\frac{P_1+P_2}{2}$) is akin to the Goldbach Conjecture: http://en.wikipedia.org/wiki/Goldbach%27s_conjecture

• Could you elaborate? @Mehrdad – Kugelblitz Mar 11 '15 at 7:18
• @Mehrdad I didn't mean 'reduce' technically; I meant to an already known problem. I do agree that it's unorthodox to use it in this way... I'll edit it. – Kugelblitz Mar 11 '15 at 7:43
• Right right....I has used 'reduced' for the lack of a better word... I'm well aware of the fact it is a weaker conjecture... I've edited it now..@Mehrdad – Kugelblitz Mar 11 '15 at 7:49

I think that it is all about prime gap.

If all natural numbers larger than 4 can be proved that it is the average of primes, the Goldbach's Conjecture will be solved, immediately