Ciro Santilli @cirosantilli 37

A conjecture is an open problem in mathematics for which some famous dude gave heuristic arguments which indicate if the theorem is true or false.

An easy to prove theorem that follows from a harder to prove theorem.

The only cases where formal proof of theorems seem to have had actual mathematical value is for theorems that require checking a very large number of case, so much so that no human can be fully certain that no mistakes were made. Some examples:

A theorem is said to be independent from a set of axioms if it cannot be proven neither true nor false from those axioms.

It or its negation could therefore be arbitrarily added to the set of axioms.

A theorem that is not very important on its own, often an intermediate step to proving something that the author feels deserves the name "theorem".