So. a^n=a problem. Today I realized that if we know it for 2^(n+1), n≠0, then it's true for 2^n+2^m for all m≠0. This is yay because this is the first infinite family of numbers I've found it to be true for (since we know it's true for 4, so it's true for 2^n+2 (even if n=0, in fact, as it's true for 3)), but it's not really a very useful generalization and while I might be able to generalize the proof, I don't think I can generalize it very far (i.e. past the case of two-bit even numbers).
(m, n represent whole numbers.)
Let n≠0 be such that for any ring R with a^(2^(n+1))=a ∀a∈R, R is commutative. Then for m≠0, given a ring R st a^(2^m+2^n)=a ∀a∈R, we have a^(2^(n+1))=a ∀a∈R, and thus R is commutative.
( Proof )
As for things not related to that stupid problem...
Firstly, the naked woman on the wall. Alex was doing her art project (this was actually the same day she took the pictures of me chewing on the soda can rings, but she hadn't decided to do that yet) and she had her charcoal with her. Sayer took a stick and used it to draw a kitten on the wall. I tried and failed to draw the Borromean rings, did it on paper, then started over on the wall, getting it right this time. I also drew the trefoil knot and the figure-8 knot.
Lisa drew a naked woman. A naked woman with a very strange-looking foot, since it overlapped my original attempt to draw the Borromean rings. Also with a rather low-detail face. Otherwise, though, an ordinary naked woman. Very - how might you say it - statuelike? - definitely not erotic, not quite even looking like a person due to the low detail on the face and the strange position she was in. Probably not something anyone would object to.
Then someone (I forget who) suggested to Lisa that her drawing hardly has any breasts and she ought to emphasize them a bit more, and also the lack of private parts (though you wouldn't be able to see those anyway). Lisa went with the first suggestion and the result has been people asking, every now and then, for the last week or so, "Who drew a naked woman on the wall?" with the definite implication that it should not be there. People seem to generally think it's ugly, it seems; I think it just looks strange. Regardless, it certainly doesn't look quite right anymore.
So apparently Lea moved out and some person by the name of Brittany has moved into her room. I didn't notice.
Truly shocking: Lucas has cut his hair. Himself. Unfortunately I did not get to see Lisa's reaction; I don't think anyone other that Lucas did. Too bad. It would be really funny to see Lisa freak out.
So the other night someone actually ripped much of the steel bar out where the lower lounge's lower ceiling turns upward. Nobody seems to know who did it. Regardless (as it was actually decided before...), apparently Housing means to actually crack down on us, i.e. Bill won't be able to get away with his constant looking the other way. If this will actually happen, who knows. How they would check on this I have no idea.
Anyway, that's all I can think of.
-Sniffnoy
(m, n represent whole numbers.)
Let n≠0 be such that for any ring R with a^(2^(n+1))=a ∀a∈R, R is commutative. Then for m≠0, given a ring R st a^(2^m+2^n)=a ∀a∈R, we have a^(2^(n+1))=a ∀a∈R, and thus R is commutative.
( Proof )
As for things not related to that stupid problem...
Firstly, the naked woman on the wall. Alex was doing her art project (this was actually the same day she took the pictures of me chewing on the soda can rings, but she hadn't decided to do that yet) and she had her charcoal with her. Sayer took a stick and used it to draw a kitten on the wall. I tried and failed to draw the Borromean rings, did it on paper, then started over on the wall, getting it right this time. I also drew the trefoil knot and the figure-8 knot.
Lisa drew a naked woman. A naked woman with a very strange-looking foot, since it overlapped my original attempt to draw the Borromean rings. Also with a rather low-detail face. Otherwise, though, an ordinary naked woman. Very - how might you say it - statuelike? - definitely not erotic, not quite even looking like a person due to the low detail on the face and the strange position she was in. Probably not something anyone would object to.
Then someone (I forget who) suggested to Lisa that her drawing hardly has any breasts and she ought to emphasize them a bit more, and also the lack of private parts (though you wouldn't be able to see those anyway). Lisa went with the first suggestion and the result has been people asking, every now and then, for the last week or so, "Who drew a naked woman on the wall?" with the definite implication that it should not be there. People seem to generally think it's ugly, it seems; I think it just looks strange. Regardless, it certainly doesn't look quite right anymore.
So apparently Lea moved out and some person by the name of Brittany has moved into her room. I didn't notice.
Truly shocking: Lucas has cut his hair. Himself. Unfortunately I did not get to see Lisa's reaction; I don't think anyone other that Lucas did. Too bad. It would be really funny to see Lisa freak out.
So the other night someone actually ripped much of the steel bar out where the lower lounge's lower ceiling turns upward. Nobody seems to know who did it. Regardless (as it was actually decided before...), apparently Housing means to actually crack down on us, i.e. Bill won't be able to get away with his constant looking the other way. If this will actually happen, who knows. How they would check on this I have no idea.
Anyway, that's all I can think of.
-Sniffnoy
