TITLE: Rigour in mathematics
QUESTION [61 upvotes]: Mathematics is very rigorous and everything must be proven properly even things that may seem true and obvious.
Can you give me examples of conjectures/theories that seemed true but through rigorous mathematical proving it was shown otherwise?

REPLY [3 votes]: Here is a case where you can easily be misled by working out some early examples.  All groups up to order 60 are solvable (a nice definition of this available on Wikipedia) -- you can show that by checking them all out. With 59 consecutive examples to start with, you might think that all groups are solvable.  But it's not true.  There is at least one group of order 60 which is not solvable.  This is directly connected to the insolubility in radicals  of polynomial of degree ≥ 5; so it would be important to get it right.
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