Sums of pairwise different prime numbers

Question

I'm interested in all natural numbers, that are the sum of pariwise different prime numbers. For example 10 is such a number, since

$10 = 7 + 3$

My suspicion is that almost every natural number is the sum of pairwise different prime numbers. Actually by my pen & paper calculations, I did not find a natural number larger than 6 that is not such a sum, but I did not calculate much, so maybe there are even small numbers that are not a sum of pairwise different primes.

So my questions are these:

Is there any natural number larger than 6, which is not the sum of pairwise different primes? If not, any idea how to prove this?

And what's about the natural numbers, that are the sum of at least n natural prime numbers, with $n \in \mathbb{N}$? Is there any knowledge about these numbers?

Regards, S. M. Roch

Edit: I'm sorry, I forgot to say for my first question, that I also accept the sum of not only two prime numbers. So for example

$20 = 11 + 7 + 2$

is also such a sum, and for every prime number the number is its sum itself.

Answer 1

A sketch of a proof:

For sufficiently large $N$, the Prime Number Theorem guarantees a prime $p$ between $N/2$ and $N-6$, $N/2\lt p\lt N-6$. This implies $6\lt N-p\lt N/2\lt N$, so strong induction tells us $N-p$ is the sum of distinct primes, $N-p=p_1+p_2+\cdots+p_n$ with $p_1\lt p_2\lt\cdots\lt p_n$. But $N-p\lt N/2\lt p$ tells us these primes are all less than $p$. Thus $N=p_1+p_2+\cdots+p_n+p$ with $p_1\lt p_2\lt\cdots\lt p_n\lt p$.

Note, Bertrand's postulate, which guarantees a prime between $N/2$ and $N$, isn't quite good enough. E.g., the only primes between $17/2$ and $17$ are $11$ and $13$, and neither $17-11=6$ nor $17-13=4$ can be written as a sum of distinct primes.

Answer 2

There are lots of numbers greater than $6$ that are not the sum of pairwise different primes, like all the odd primes greater than $5$ that are not the larger part of a twin prime pair. If $n$ is odd, then so is $n - 2$. Remember that two odd primes add up to an even number.

In the case of $11$, we see that $11 - 2 = 9$, $11 - 3 = 8$, $11 - 5 = 6$ and $11 - 7 = 4$. Clearly $9$ is the square of an odd prime, and the rest are even numbers other than $2$. Look at Sloane's A014092 for plenty more examples.

For even $n$, you're talking about the Goldbach conjecture, as the commenters have already mentioned. For odd $n$, there is what is sometimes called the "weak" Goldbach conjecture, that every sufficiently large odd $n$ can be represented as the sum of three odd prime numbers.

Mathematicians have been able to prove both versions of the Goldbach conjecture for some numbers so large they are almost beyond the reach of our computers. But that doesn't disprove the possibility that a counterexample could exist just a little beyond your computer's ability.