The weak Goldbach conjecture, that all odd numbers greater than n can be written as a sum of at most m prime numbers, for which the most difficult statement is n=3 and m=3, is an interesting problem because it can be (and is being) gradually chipped away at with published proofs for larger n or m.
Is there anything "meta" interesting about the sequence of proofs toward smaller and smaller n and m? A measure of "difficulty" of the proof? Can a machine learn how to prove simple then difficult things? Can we somehow extrapolate a sequence of valid proofs to create a new valid proof?
The Goldbach conjecture is tantalizing because a typical large odd number can be expressed as a sum of 3 primes in a huge number of ways, but the conjecture merely asks for the existence of one. Any other conjectures have this feeling of overkill?
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