14. INTEGERS
Explanation:
z = x − 1
Solution:
The equation is equivalent to (x - z) (x + z) + y\(^{\1}\)= n. If we set x - z = then we obtain 2x - 1 + y\(^{\1}\) = n and \(\text{x} = \frac{\text{n + 1 - y}^2}{2}\) . Now, it suffices tso take y = n + m, where m is an odd integer to insure that x is an integer as well. Indeed, if m = 1 kt 1 then,
\(\) \(\frac{\text{n + 1 - y}^2}{2} = \frac{\text{n+1-(n+2k+1)}^2}{2}=-\frac{\text{n (n + 1)}}{2}\\-2\text{nk}-2\text{nk}^2-2\text{k,}\)
it is also an integer number. obviously an integer. Since z = x - 1 z = x - 1