Are there any counterexamples to a generalized Goldbach conjecture?

Question

The Goldbach conjecture suggests that every even integer greater than $2$ can be expressed as a sum of two primes. For example:

$$10 = 5+5$$ $$12 = 7+5$$ $$14 = 7+7$$

What is so special about even integers? Can we generalize the conjecture as follows?

Every integer multiple of $n$ greater than $n$ can be expressed as a sum of $n$ primes.

For example:

$$15 = 3+5+7$$ $$21 = 7+7+7$$ $$27 = 7+7+13$$

In the above example, I didn't include $24$, because I'm adding the additional restriction that the number is not a multiple of a prime less than $n$.

Answer 1

I don't think this is a generalization of Goldbach, since it can be proved rather easily if Goldbach is assumed to be true.

Fix your whole number $n \ge 3$. Clearly $2n$ is a sum of $n$ primes: $$2n = \underbrace{2 + 2 + \cdots + 2}_\text{$n$ times}$$

If $k \ge 3$ you can write $$kn = \underbrace{2 + 2 + \cdots + 2}_{\text{$n-2$ times}} + 2m$$ if $kn$ is even, and $$kn = \underbrace{2 + 2 + \cdots + 2}_{\text{$n-3$ times}} + 3 + 2m$$ if $kn$ is odd. According to Goldbach, in each case $2m$ can be written as a sum of two primes so that $kn$ is a sum of $n$ primes.

Answer 2

Goldbach's actual conjecture was "Every integer greater than $5$ is the sum of 3 primes" (in particular, $24 = 2+11+11$). The one everyone talks about, and the one you heard, is a refinement that is equivalent. To see this, note that, since $2$ is the only even prime, every even sum of 3 primes greater than $2+2+2=6$ must contain exactly 1 addend of $2$. Thus, if Goldbach was right, every even number greater than $4$ is the sum of 2 prime numbers, and $4$ itself is $2+2$. Conversely, every odd number is $3$ greater than an even number while every even number is $2$ greater than an even number, and both $3$ and $2$ are themselves prime.

Moreover, every number is $2$ greater than another number, and $2$ is prime, so, if Goldbach was right, then every integer greater than $7$ is a sum of 4 primes, every integer greater than $9$ is the sum of 5 primes, every integer greater than $11$ is the sum of 6 primes, and so on ad infinitum, giving us the "endless" Goldbach conjecture: every integer greater than $2n-1$, for a suitably large $n$, is the sum of n primes. Since $2n-1 < 2n$ for all $n$, it follows that your conjecture is a weaker corollary of Goldbach's, as you limited yourself to multiples of $n$, and further to those that did not have a prime factor smaller than $n$.

But then, why does the refined Goldbach conjecture, the one that started this mess, limit itself in the same way? Because $2$ is a pain. As the only even and smallest prime, it's the edge case to end all edge cases. Number theory proofs often have three parts: powers of $2$, powers of primes other than $2$, and all other numbers. And the first part is usually the longest, and sometimes has extra conditions. For Goldbach, the fact that $2$ is the only even prime and that not all odd numbers are prime means that there are some odd numbers that cannot be the sum of 2 primes. The smallest example is $11$, as $9= 3\times 3$. As for the evens, well, that's the million dollar question, now isn't it?

All this to say: any counterexamples to your conjecture will be counterexamples to Goldbach, and we haven't found any.