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 $p^{n-1+ord_p([n/p]!)}$, where $ord_p(m)$ denotes the largest integer $\alpha$ such that $p^{\alpha}$ divides $m$.Comment: 20 page

Polynomial extension of Fleck's congruence

Let $p$ be a prime, and let $f(x)$ be an integer-valued polynomial. By a combinatorial approach, we obtain a nontrivial lower bound of the $p$-adic order of the sum $$\sum_{k=r(mod p^{\beta})}\binom{n}{k}(-1)^k f([(k-r)/p^{\alpha}]),$$ where $\alpha\ge\beta\ge 0$, $n\ge p^{\alpha-1}$ and $r\in Z$. This polynomial extension of Fleck's congruence has various backgrounds and several consequences such as $$\sum_{k=r(mod p^\alpha)}\binom{n}{k} a^k\equiv 0 (mod p^{[(n-p^{\alpha-1})/\phi(p^\alpha)]})$$ provided that $\alpha>1$ and $a\equiv-1(mod p)$.Comment: 11 page