If you have k numbers that add up to zero (with possible plus or minus signs), what is the maximum number ‘n’ of prime factors of the number with the least prime factors — what kind of inequality can we say about ‘n’ and ‘k’? The primes can be repeated, as long as there are no common factors for each term.
For example, 2*5*29 + 7*7*11 – 2*7*23 – 3*13*13 == 0 is true. In this case, n = 3 and k = 4. [In this case, the 12 primes are divided into four equal groups of 3, but that is not a requirement in general.]
MOTIVATION: the push and pull of multiplication and addition is an interesting topic. This seems a question in the same vein as Fermat’s Last theorem or the ABC conjecture, and it seems like a very natural similar question to ask. I suspect we’re not even close to being able to answer it, but would like to know if there’s a conjecture regarding ‘n’ and ‘k’, especially as they get larger, similar to the ABC conjecture inequality.