Carmichael's Conjecture PROVEN (Carmichael's Conjecture states that the value of the Euler phi function, over the domain of all possible integers, is never unique)
Just to be clear, this is Carmichael's Totient Function Conjecture, which as far as I know hasn't yet been proven, and not the other Carmichael's Conjecture, which was proven in 1994 by Alford, W. R.; Granville, A.; and Pomerance, C. in ``There Are Infinitely Many Carmichael Numbers.'' Ann. Math. 139, 703-722, 1994.
The proof proposed by Filip Saidak in a letter to his teacher in 1997 was found to be flawed. See A discussion of this and other recent progress on famous math problems.