Is every prime the average of two other primes?

$\forall {p_1\in\mathbb{P}, p_1>3},\ \exists {p_2\in\mathbb{P},\ p_3\in\mathbb{P}};\ (p_1 \neq p_2) \land (p_1\neq p_3) \land (p_1 = \frac{p_2+p_3}{2})$

Now I'm not a 100% sure about this, but I vaguely remember proving this once, but I cannot recall how I did it right now.

It's also a bit like a weaker version of Goldbach's conjecture, where now those even numbers that a double a prime are the sum of two primes, with the added condition of the summed primes being different.

So I'm asking if someone can provide/link to a proof of this? Because I've been looking around wikipedia and google but I cannot find this statement anywhere.

Solution 1:

This conjecture is very similar to Goldbach's conjecture, namely that every even number is the sum of two prime numbers. Here you are claiming that in the particular case that that even number is twice a prime number $p>3$, there is at least a second way to write it so (the first one being $p+p$). Goldbach's conjecture has resisted so far (a very long time) to attempts to prove it, even with sophisticated number theoretic means; my guess is that the conjecture asserting a second expression for numbers of the form $2p$ is not substantially easier to settle than Goldbach's conjecture (unless it happens to be false).

What you can say more precisely is that if Goldbach's conjecture and this conjecture are both true, then so is the somewhat stronger version of Goldbach's conjecture"

Every even number $n>6$ is the sum of two distinct primes.

Of course those are two big "if"s.

Solution 2:

You can make a more expansive conjecture with the same degree of confidence:

Every integer greater than 3 can be expressed as the average of two primes.

If a number is the average (or difference) of two primes, by doubling the number it has a partition of those two primes. So, for example, $(7+31)/2=19$ becomes $7+31=2∗19$. The Goldbach conjecture applies to even numbers only, but the average of two primes applies to every number - even, odd, prime - bigger than 3.

CSV of first 100,000: int, diff, p1, p2, type

Solution 3:

The Goldbach conjecture says that every even integer >= 4 is the sum of two primes.

Now quite obviously if p is a prime, then the even integer 2p is the sum of two primes, and if p is not a prime, then 2p is not the sum of two equal primes. We can remove the trivial cases 4 and 6. So I'll write the trivially equivalent modified Goldbach conjecture: Every even integer >= 8 that is not twice a prime is the sum of two different primes.

We now see that the conjecture that is discussed here has nothing to do with the Goldbach conjecture, because it says: Every even integer >= 8 that is twice a prime is the sum of two different primes. It discusses exactly those even integers that the modified Goldbach conjecture isn't concerned about.

(On the other hand, if a proof for my modified Goldbach conjecture is found, it is unlikely that it would rely on the fact that n/2 is composite or use it in any way, so quite likely a proof for the modified Goldbach conjecture would prove this conjecture as well.