Consider the equation .

Show that all non-integral solutions of this equation lie on exactly two lines.

- May 20th 2009, 09:08 PMfardeen_genShow that all non integral solutions lie on exactly two lines
Consider the equation .

Show that all non-integral solutions of this equation lie on exactly two lines. - May 21st 2009, 04:17 AMHallsofIvy
The left hand side

**is**an integer whether x and y are or not. If x and y are not integer, then the "fraction parts" must cancel. Either x= a+ r, y= b- r or x= a- r, y= b+ r, where a and b are integers, 0< r< 1. Those give the two lines. (And, of course, ab= a+b.) - May 21st 2009, 05:47 AMTheAbstractionist
- May 21st 2009, 05:52 AMpankaj
and

- May 21st 2009, 06:01 AMIsomorphism
- May 21st 2009, 11:31 AMpankaj
[x][y]=x+y

[x][y]=[x]+[y]+{x}+{y} ({x} and {y} denote the fractional parts of x and y respectively)

0 {x}+{y}<2

0 ([x][y]-[x]-[y])<2

1 (1+[x][y]-[x]-[y])<3

1 ([x]-1)([y]-1)<3

([x]-1)([y]-1)=1,2

__Case 1__

([x]-1)([y]-1)=1

[x]-1=1 and [y]-1=1

[x]=[y]=2

{x}+{y}=(2)(2)-2-2=0

{x}={y}=0,but then and will be integers

__Case 2__

{x}+{y}=0-0-(-1)=1 and thus

__Case 3__

{x}+{y}=0-0-0=0

{x}={y}=0,but then and are integers

__Case 4__

{x}+{y}=6-2-3=1

Therefore,