A moment of inertia
Jul. 25th, 2005 06:50 pmSo today in Zeta Rohrlich finished proving the Kummer congruence (or a Kummer congruence?). After that, he just kind of paused and staood there as the class absorbed the proof. There's a term he's borrowed from physics that he likes to use to describe this sort of pause, he says - a moment of inertia. "Instead of calculating moments of inertia, let's have a moment of inertia."
Thing I forgot to include before: So today during number theory Glenn started talking about which numbers were QRs mod primes, and asked for conjectures. Jason Bland - just from looking at tables of QRs he had compiled - came up with the amazing conjecture that -3 is a QR mod p iff p≡1 mod 6. Yeah... that was just wow.
Also, today was the "Partners in Prime" pset. Some noticeable things: Eli and Jason got grouped together on 101, the least one, which has a primitive root of 2. I guess they figured those two have better things to do with their time. Also, Victoria, Irina, and Infinity-boy got grouped together, and so did 3 people whose last names all started with 'W'.
Yay, actual significant progress on lab today! So I'm trying to prove the periodicity of x choose k (x varying) mod pr, and have been for quite some time. I already know quite well what it is, the data makes it pretty clear, but I've been trying to find a proof and not really getting anywhere. So the other day Man-Yu comes to work on it as well. Now, Man-Yu, so far as I can tell, has done pretty much nothing for the entire lab. He asks, what about using Vandermonde's Convolution? That looks nasty, I said. You go try that if you really think it'll help. Well, today I realized none of my approaches were working and I figured, hey, why not try Vandermonde's Convolution. And, amazingly enough, Man-Yu's idea worked! It didn't actually prove my conjecture in general, but it as an upper bound, which with what I already have proves it for primes. And, of course, I ended up getting this before Man-Yu did. :P
UPDATE: Actually, extending this a bit further, I have a full proof of the conjecture! Much yayness!
-Sniffnoy
Thing I forgot to include before: So today during number theory Glenn started talking about which numbers were QRs mod primes, and asked for conjectures. Jason Bland - just from looking at tables of QRs he had compiled - came up with the amazing conjecture that -3 is a QR mod p iff p≡1 mod 6. Yeah... that was just wow.
Also, today was the "Partners in Prime" pset. Some noticeable things: Eli and Jason got grouped together on 101, the least one, which has a primitive root of 2. I guess they figured those two have better things to do with their time. Also, Victoria, Irina, and Infinity-boy got grouped together, and so did 3 people whose last names all started with 'W'.
Yay, actual significant progress on lab today! So I'm trying to prove the periodicity of x choose k (x varying) mod pr, and have been for quite some time. I already know quite well what it is, the data makes it pretty clear, but I've been trying to find a proof and not really getting anywhere. So the other day Man-Yu comes to work on it as well. Now, Man-Yu, so far as I can tell, has done pretty much nothing for the entire lab. He asks, what about using Vandermonde's Convolution? That looks nasty, I said. You go try that if you really think it'll help. Well, today I realized none of my approaches were working and I figured, hey, why not try Vandermonde's Convolution. And, amazingly enough, Man-Yu's idea worked! It didn't actually prove my conjecture in general, but it as an upper bound, which with what I already have proves it for primes. And, of course, I ended up getting this before Man-Yu did. :P
UPDATE: Actually, extending this a bit further, I have a full proof of the conjecture! Much yayness!
-Sniffnoy
