In this paper, the survivable routing problem on elastic optical networks (EONs) for the single link-failure case is studied. The failure-independent path-protecting p-cycles (FIPP p-cycles) protection scheme is used for lightpath protection. For the given network and the set of connection requests, the lightpaths used to route the connection requests and the protecting cycles are found. In this paper, two new properties of FIPP p-cycles are introduced on EONs to improve the spectrum efficiency. Several heuristic algorithms are proposed to solve this problem and simulations are run in the static case to minimize the total number of frequency slots. The performance of fragmentation ratio and resource utilization ratio are also examined.