The fractional quantum Hall effect (FQHE) is observed when a two-dimensional system of electrons is subjected to a high perpendicular magnetic field. It exhibits unique properties such as non-trivial topology, quasiparticles with fractional charges and fractional braiding statistics. The theory of composite fermions (CFs) plays a pivotal role in the understanding of FQHE.

One of the notable states that captured significant attention is the 5/2 state in GaAs quantum wells, which is the half-filled second Landau level. Various paired states of CFs, e.g., Moore Read Pfaffian or the parton states, have been proposed for the 5/2 state, but their connection to the underlying composite fermion Fermi sea is unclear. A formulation analogous to the conventional Bardeen-Cooper-Schrieffer theory for electrons is lacking. We address this by constructing a BCS paired state of CFs with two variational parameters and find a p-wave pairing instability at 5/2, consistent with experiments and other theoretical predictions. We extend our approach to understand the pairing channel for the even-denominator FQHE state observed at half-filling in the n=3 Landau level of graphene, where we find an f-wave paired state. We also investigate the origin of FQHE at 1/2 and 1/4 fillings in wide GaAs quantum wells. We find a p-wave paired state at 1/2 filling, and an f-wave paired state at 1/4 is stabilized at large well widths and densities. The theoretical phase diagram is in excellent agreement with the experiments. Our work may inspire further investigations of these states through thermal Hall and Hall viscosity measurements.