In this article, we study the set of parameters c ∈ C for which two given complex numbers a and b are simultaneously preperiodic for the quadratic polynomial f_c(z) = z² + c. Combining complex-analytic and arithmetic arguments, Baker and DeMarco showed that this set of parameters is infinite if and only if a² = b². Recently, Buff answered a question of theirs, proving that the set of parameters c ∈ C for which both 0 and 1 are preperiodic for f_c is equal to {-2, -1, 0}. Following his approach, we complete the description of these sets when a and b are two given integers with |a| not equal to |b|.

The author would like to thank his Ph.D. advisors, Xavier Buff and Jasmin Raissy, for helpful discussions without which this paper would not exist and the anonymous referee for his comments.