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