Given a ring with multiplicative identity, let denote the set of its units.
What is the smallest positive integer such that there does not exist any finite ring such that contains exactly elements?
the answer is you don't need to assume is finite. it's clear that for there exist rings with now let be a ring and suppose then in because
otherwise would be a subgroup of which is impossible because is odd. now look at the group algebra by Maschke's theorem is semisimple and thus by Wedderburn-Artin
theorem where each is a division ring containing (here means the ring of matrices with entries from ) since is finite, each is
finite. but we know that a finite division ring is a field. so each is a finite field extension of i.e. for some
also if for some then which is odd only for thus but then and
hence: which is clearly impossible. Q.E.D.
Remark: the above solution uses some powerful theorems in ring theory and so you should expect to get a lot more than just answering your question out of it. so ... see if you can do that!
Over in the AoPS/MathLinks forum at the moment, some people are trying to determine if there exists a finite ring with unity such that The latest development appears to be that if such a ring exists, it must have characteristic 4. It looks exciting – I wish I knew more ring theory so I could join in and not be left out of the fun.