Determine the set of values of \(n\) for which \(\frac { n ^ { 2 } - 1 } { 2 }\) and \(\frac { n ^ { 2 } + 1 } { 2 }\) are positive integers.
A 'Pythagorean triple' is a set of three positive integers \(a , b\) and \(c\) such that \(a ^ { 2 } + b ^ { 2 } = c ^ { 2 }\).
Prove that, for the set of values of \(n\) found in part (a), the numbers \(n , \frac { n ^ { 2 } - 1 } { 2 }\) and \(\frac { n ^ { 2 } + 1 } { 2 }\) form a Pythagorean triple.