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While the one-Cooper pair problem is now a textbook exercise, the energy oftwo pairs of electrons with opposite spins and zero total momentum has not beenderived yet, the exact handling of Pauli blocking between bound pairs being notthat easy for N=2 already. The two-Cooper pair problem however is quiteenlightening to understand the very peculiar role played by the Pauli exclusionprinciple in superconductivity. Pauli blocking is known to drive the changefrom 1 to $N$ pairs, but no precise description of this continuous change hasbeen given so far. Using Richardson procedure, we here show that Pauli blockingincreases the free part of the two-pair ground state energy, but decreases thebinding part when compared to two isolated pairs - the excitation gap to breaka pair however increasing from one to two pairs. When extrapolated to the denseBCS regime, the decrease of the pair binding while the gap increases stronglyindicates that, at odd with common belief, the average pair binding energycannot be of the order of the gap.

Two-Cooper-pair problem and the Pauli exclusion principle