A number-theoretic approach to homotopy exponents of SU(n)

We use methods of combinatorial number theory to prove that, for each $n>1$ and any prime $p$, some homotopy group ${\pi }_i(SU(n))$ contains an element of order $p^{n-1+or{d}_p([n\mathrm{/}p]!)}$, where $or{d}_p(m)$ denotes the largest integer $\alpha$ such that $p^{\alpha }$ divides $m$.