 TEACHER: I agree with you that the conjecture has received a severe criticism by Alpha’s counterexample. But it is untrue that the proof has “completely misfired”. If, for the time being, you agree to my earlier proposal to use the word “proof” for a “thought-experiment which leads to decomposition of the original conjecture into subconjectures”, instead of using it in the sense of a “ guarantee of certain truth”, you need not draw this conclusion. My proof certainly proved Euler’s conjecture in the first sense, but not necessarily in the second. You are interested only in proofs which “prove” what they have set out to prove. I am interested in proofs even if they do not accomplish their intended task. Columbus did not reach India but he discovered something quite interesting.