"Any positive integer can be expressed as the sum of x primes, where x is a factor of the integer that is not equal to either the integer or 1."

Or alternatively:

"The product of any positive integers that are greater than 1 can be expressed as the sum of x primes, where x is any of the positive integers."

For example, 25 = 5 * 5. The new conjecture would have it that 25 can be expressed as the sum of 5 primes. 25 can in fact be expressed in this way: 25 = 2 + 2 + 3 + 7 + 11.

Example: 26 = 2 * 13. 26 = 13 + 13. Also, 26 = 2 + 2 + 2 + 2 + 2 + 2 + 2 + 2 + 2 + 2 + 2 + 2 + 2.

Example: 28 = 4 * 7. 28 = 7 + 7 + 7 + 7. Also, 28 = 2 + 3 + 3 + 3 + 3 + 3 + 11.

Are there any obvious violations of this extension?

If not, can this extension be proven assuming that Goldbach's conjecture is true? The reverse is obviously the case since Goldbach's conjecture is simply the specific case for the multiples of 2 greater than 2.

Robert Burrage
goak@airmail.net