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A "hard" question about the number 10...?
Actually, I have no clue whether this is hard or not...
Prove or disprove:
10 cannot be written as the difference of squares of rational numbers.
Awww... bummer.
Oh well, back to the drawing board then (I was so focused on trying to prove it that I missed the obvious @_@)
... maybe I've forgotten what a minimal polynomial is... heheh...
Actually, thanks for the responses, guys. I'd been trying to prove a polynomial was minimal. My work led me to this statement, but it didn't seem to work (now I see why it wasn't).
I just looked back at my work and realized I have a stronger assumption I can make that changes the problem to the statement that √2 and √3 are irrational, which is easy.
Again, thanks for the answers, guys.
3 Answers
- 1 decade agoFavorite Answer
x^2-y^2=10 => (x-y)(x+y)=10 => here we just guess that 2*5 is 10, so
x-y=2
x+y=5
here we get x=3.5, y=1.5
so i prooved..
quite simple, huh?
by the way, the same would be if we take 4*2.5=10
x-y=2.5
x+y=4
x=3.25 y=0.75
unsing this way, you can get more...
Source(s): kolibrizas@gmail.com for more discussion :) - SteinerLv 71 decade ago
Let b ne any positive integer.
x² - y² = n
(x + y)(x - y) = n
A possible solution is
x + y = n
x - y = 1
This gives
x = (n +1)/2
y = ( n-1)/2
Siince n is an integer, x and y are rational. Hence, every positive integer (actually, every integer) can be given by the difference of 2 squares of rational numbers.
- Anonymous1 decade ago
no idea but i do know n! cannot be written as the difference of two integer squares
