Trying to negate a statement for a proof by contradiction

To prove that a statement R is true contradiction, my textbook says to begin by assuming that R is false (http://imgur.com/a/yGlRS) or by assuming the negation of the statement, i.e. ~R.

I'm currently trying to prove the statement: For every integer m such that 2|m and 4∤m, there exist no integers x and y for which x^2 +3y^2 =m.

I tried to negate this statement and came up with:

There exists an integer m such that 2∤m and 4|m and x^2 +3y^2≠m for all integers x and y.

However, in the textbook it apparently should be:

There exist an integer m such that 2|m and 4∤m and integers x and y for which x^2 +3y^2 =m

I tried to take each part of the statement and negate it, e.g. "For every" becomes "there exists" and vice versa, also "=" becomes "≠" etc. Can someone please explain to me what I'm doing wrong, and how you get to the answer in the textbook?