User blog comment:Vel!/Call for math facts/@comment-1605058-20140730192227


 * Knuth has suggested that Goldbach's conjecture might be a problem independent of our axioms. If that's turns out to be correct, then we will know that the conjecture is correct anyways - a disproof would take finite time.


 * Steiner-Lehmus theorem states that if two angle bisectors in a triangle have the same length, then triangle is isosceles. Even though fact itself is really elementary, no simple proof of it is known. John Conway has shown that method of  what he calls "equality-chasing", meaning that we just use equalities and congruences of triangles, is very unlikely to exist.