Wednesday, May 31, 2006

I'll stop writing in Chinese for the time being

English translation and more below.






For the time being, I’ll be less diligent in my Chinese writing. At the moment, the most important aspect of my education is to vastly improve my vocabulary and be familiar with the grammar and sentence structures. To this end, I will be reading and looking up the dictionary; writing is of no help now.

I feel I’ve made decent progress over the past weeks. When I started I had trouble recognising simple characters like … I’ll not embarrass myself by publishing that sample list here. I can now read news articles, although not to absolute comprehension.


On a separate note, I’ve unexpectedly got 2 very interesting questions on my hands, one geometry, the other higher arithmetic.

The higher arithmetic one is particularly fascinating (because it can be written succinctly):

a raised to the power of b is congruent to 1 modulo (a-1) for all a and b that are natural numbers

It’s a generalised statement on one of the key arguments used in solving the JFE8555 problem. Previously, it had never been rigorously proven in solving that problem, although the numerical examples gave (me) ample confidence in the equation.

It’s a two parameter problem, so proof by induction is not likely to work. It’s going to be difficult.

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Blogger 小李飞刀 said...

This comment has been removed by a blog administrator.

2:28 pm, May 31, 2006  
Blogger 小李飞刀 said...

Hmm.. I think I just proved your theorem by induction.

*spoiler warning*

(a^(n+1)-1)= (a-1)(a^(n)+1)+a^(n)-a
= (a-1)(a^(n)+1)+a(a^(n-1)-1)

which is divisible if (a^(n-1)-1) is divisible by (a-1).

Since index is 2 numbers behind, have to show for first two cases to prove for all of N.

Since (a-1)/(a-1)=1 and
(a^2-1)/a-1=a+1, it is therefore the case. QED maybe?

2:33 pm, May 31, 2006  
Blogger Lao Chen said...

Oh, nice. I thought induction would not work, turns out one variable is a completely free variable, and the other one can be sorted out using induction. Pretty.

9:16 pm, May 31, 2006  

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