Let n be a positive integer and S_{n}=\{1,2, \ldots, 2 n-1,2 n\}. A perfect pairing of S_{n} is defined to be a partitioning of the 2 n numbers into n pairs, such that the sum of the two numbers in each pair is a perfect square. For example, if n=4, then a perfect pairing of S_{4} is (1,8),(2,7),(3,6),(4,5). It is not necessary for each pair to sum to the same perfect square.
(a) Show that S_{8} has at least one perfect pairing.
(b) Show that S_{5} does not have any perfect pairings.
(c) Prove or disprove: there exists a positive integer n for which S_{n} has at least 2017 different perfect pairings. (Two pairings that are comprised of the same pairs written in a different order are considered the same pairing.)