A "hard" question about the number 10...?

Question

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.

kolibrizas · Accepted 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...

Steiner · Answer

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.

Anonymous · Answer

no idea but i do know n! cannot be written as the difference of two integer squares

A "hard" question about the number 10...?

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