Suddenly, my keyboards for Korean does not work. So today I wrote my today's diary in English.
- What I learned today
Risk is manageable
Take Risk and avoid Danger!
2 . Pythagorean primitive triples [In the process of reading quiz materials]
Theorem[diophantus] A triple of integers is a primitive Pythagorean triple if and only if it may be written in the form (m^2-n^2, 2mn, m^2+n^2) or (2mn, m^2 -n^2, m^2+n^2), where m>n are relatively prime positive integers of different parity.
Here different parity means if m is even then n is odd and vice versa.
How one can prove this theorem?
One simple method is starting from x^2+y^2 = z^2 and rearranging until
1 = (z/x + y/x)(z/x-y/x)
--> z/x+y/x = m/n, z/x - y/x = b/a
--> solve above two equations for z/x and y/x
--> z/x = (a^2+b^2)/2ab , y/x = (a^2-b^2)/2ab
Note that if we assume b<a, since x,y, and z have no common factor, if x is odd and a and b are relatively prime, x=ab so both a and b are odd, y=(a^2-b^2)/2, and z = (a^2+b^2)/2
3 . Typesetting is too boring and difficult...
But I have a due date to finish.
4 . Preparing My mother's birthday.
5 . what happen to my keyboard?
The keyboard and speaker in my laptop seem not working... Do I have to reset? or replacing new laptops?
To my knowledge,
Risk is manageable because it is not realized yet.
Risk could be turned to be danger in the end in case it is not expected or not well treated.
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