TITLE: Theorems demoted back to conjectures
QUESTION [63 upvotes]: Many mathematicians know the Four Color Theorem and its history: there were two alleged proofs in 1879 and 1880 both of which stood unchallenged for 11 years before flaws were discovered.
I am interested in more such examples, especially, in former (famous or interesting) "theorems" where no remedy could be found to the present day.

REPLY [11 votes]: Carmichael's totient function conjecture: 

For every positive integer $m$, there is at least one integer $n \neq m$ such that $\varphi(m)=\varphi(n)$. 

Carmichael stated this as a theorem in 1907, but he retracted his proof in 1922 and stated this conjecture as an open problem.