Saturday, August 28, 2004
... wat a week... so many things happening con currently... haha not all of which are good of course... including something at home... haha oh well... that shld be resolved rather efficiently... albeit with a loss of a certain amt of cash... haha hmmm dunno y mum wuz so uptight abt it... anywayz... xtine's in hospital now... kidney infection i believe... hope she gets well soon... shall vist her with gerri and francis tmr... evening... which ought to be a fun day considering i did work today... haha the words nicholas and work dun seen to like each other very much... haha
hmmm anyway, as part of nicholas' collection of nice proofs, i shall relate an interesting thing that happened on friday during 4263 lecture... haha INTRO TO ANALYTIC NUMBER THEORY... basically the class wuz and still is small... say 14-16 pple only... so the lecturer CHH sed that he wanted to get to know us better during the tutorial... so he called some of us up to present questions from the tutorial... he had in his hand... a class list with our pictures on it... so he wuz busy trying to fit names to faces... haha... then... came qn4... he asked the class "who would like to present qn 4?"... no reply... he asked a few students sitting near him if they would like to try and they all sed they din noe how to do it.... then came the interesting bit... he sighed... and sed "aiyah... then i haf to use my SECRET WEAPON!!"... and then... "KJ, can u pls present qn 4 on the board." hahaha i wuz momentarily stunned... haha but i already knew the sheer intelligence this guy possessed man... but i really din noe how famous he wuz in the entire dept... this guy really has talent man... GENIUS seems to be an understatement... haha AND... he actually went up to the board without and answer and did the whole question on the spot!! my god!! i nearly wanted to kowtow to him there and then!! it wuz sooooo cool!! he really does have the "seh" and the poise of a genius!!... haha
anyway here is wat he proved:
Suppose p is prime, then (p-1)!+1 is a p power if and only if p=2, 3 or 5.
Proof. We need to use the following lemma which can be easily proved: If (n-1)² (n^k-1), then n-1 k. Let us first assume that (p-1)!+1 = p^k for some positive integer k, which is (p-1)! = p^k -1. Now let use assume that p prime and p ≥ 7. Now we already know that (p-1) (p-1)!, but in order to use the other lemma we need another factor of (p-1). Consider this: since p>2 and p is prime, then p-1 must be even. That means that (p-1)/2 is an integer and since p>5, then (p-1)/2 >2. Thus, since (p-1)/2 and 2 are 2 values in the factorial of (p-1). That is, (p-1)! = (p-1)(p-2)...((p-1)/2)...(3)(2)(1). which means that (p-1)((p-1)/2)(2) (p^k - 1) so according to the lemma above, p-1 k which implies that p-1 ≤k and p-2 ≤k-1. Thus we arrive at the following: p^k-1>p^k-p^(k-1)=p^(k-1) (p-1)≥p^(p-2) (p-1)≥(p-2)^(p-2) (p-1)≥(p-2)!(p-1)=(p-1)! which is a contradiction to our original assumption. The converse is easy to check and that completes out proof.
And there u have it... haha one of another nice nice proofs... haha hmmm speaking of which i wuz browsing thru the prospectus of the university of london, imperial college... thinking of going there for my graduate studies to get my MSc from there... then i realised that if i want to do algebra for my paper, i shld actually apply to Queen Mary... apparently it has a maths dept that specialises in algebra!!! haha how cool is that... but then again.. i need a really good second uppers degree in order to get in and succeed... haha i really hope i do well this sem... ok ok time to go do something abt that... haha promise to write more tmr... haha ciao...
FIRE!!
finally back on track... a bit slow... but tau huey is a good reward for keeping up with it...
{9:00 PM}
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