Sharpness of exponent bounds for SU(n)SU(n)

The pp-primary v1v1-periodic homotopy groups of a topological space XX, denoted by v1−1π∗(X)(p), are roughly the parts of the homotopy groups of XX localized at a prime pp which are detected by KK-theory. We will use combinatorial number theory to determine, for pp an odd prime, the values of nn for which v1−1π2(n−1)(SU(n))(p)≅Z/pn−1+νp(⌊np⌋!). As a corollary, we obtain new bounds for the pp-exponent of π∗(SU(n))π∗(SU(n)).