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"The physicists defer only to mathematicians, and the mathematicians defer only to God ..." - Leon M. Lederman
Euler also conjectured that Orthogonal Latin Squares don't exist for 4n + 6 for every n. This is false, and while the counter example is not in the size of the millions, latin squares get very difficult to find for any number greater than 5 or so.
"In the real world, this would be a problem. But in mathematics, we can just define a place where this problem doesn't exist. So we'll go ahead and do that now..."