> why not numbers that can only be produced by adding 3 primes together?
Goldbach's weak conjecture: Every odd number greater than 5 can be expressed as the sum of three primes.
First proposed in 1742, and proven in 2013 [0]. The original proposal considered even numbers as well, nowadays those are covered by Goldbach's strong conjecture, with a tighter bound of 2 primes.
> why not only numbers that can be produced by multiplying squares greater than 1?
You mean squares containing at least 2 distinct prime factors? Fully classifying this set of integers would fit well on an undergrad intro to proofs exam.
the actual examples I give are just that, examples. why not numbers that can only be produced as the sum of 17 primes? or 459? or numbers that have the same number of factors as their digits added together does? there are infinite of these constraints that can be invented. why is this one particularly interesting
I can't tell if you're really asking, so I'll answer as if you are:
learning these things brings greater understanding and greater understanding brings advances in multiple fields.
the most fundamental questions are going to be answered (or at least conceived of and worked on) first. answering the most fundamental questions often brings the largest leaps in understanding as well, which is why untouchable numbers are considered more notable than "numbers which can be expressed as the sum of 17 primes", and so on.
As an attempt - it's a simple definition, easy to check for small numbers, but leads to some difficult questions - are there infinitely many of them? (found yes by Erdos) How dense are they in the number line? Are there any odd ones > 5? "Numbers that can only be produces as the sum of 459 primes" is not as easy to find examples of, so less aesthetically pleasing to mathematicians.
Goldbach's weak conjecture: Every odd number greater than 5 can be expressed as the sum of three primes.
First proposed in 1742, and proven in 2013 [0]. The original proposal considered even numbers as well, nowadays those are covered by Goldbach's strong conjecture, with a tighter bound of 2 primes.
> why not only numbers that can be produced by multiplying squares greater than 1?
You mean squares containing at least 2 distinct prime factors? Fully classifying this set of integers would fit well on an undergrad intro to proofs exam.
[0] https://arxiv.org/pdf/1501.05438.pdf