Particle-particle propagator in the algebraic diagrammatic construction scheme at third order

The algebraic diagrammatic construction scheme at nth order includes all Feynman diagrams up to nth order and a partial summation of diagrams arising from the latter to infinite order. In the present work this scheme is carried out for the first time at third order for the particle-particle propagator. Because of the many diagrams involved in third order the approach requires a very careful analysis of terms which describe the corresponding perturbation expansion of the propagator. The relevant technical details of the analysis are presented.