Prime power divisors of binomial coefficients

AbstractIt is known that for sufficiently large n and m and any r the binomial coefficient (nm) which is close to the middle coefficient is divisible by pr where p is a ‘large’ prime. We prove the exact divisibility of (nm) by pr for p > c(n). The lower bound is essentially the best possible. We also prove some other results on divisibility of binomial coefficients

Q-Series with Applications to Binomial Coefficients, Integer Partitions and Sums of Squares

In this report we shall introduce q-series and we shall discuss some of their applications to the integer partitions, the sums of squares, and the binomial coefficients. We will present the basic theory of q-series including the most famous theorems and rules governing these objects such as the q-binomial theorem and the Jacobi’s triple identity. We shall present the q-binomial coefficients which roughly speaking connect the binomial coefficients to q-series, we will give the most important results on q-binomial coefficients, and we shall provide some of our new results on the divisibility of binomial coefficients. Moreover, we shall give some well-known applications of q-series to sums of two squares and to integer partitions such as Ramanujan’s modulo 5 congruence

v1-Periodic 2-exponents of SU(2^e) and SU(2^e + 1)

We determine precisely the largest v1-periodic homotopy groups of SU(2^e) and SU(2^e + 1). This gives new results about the largest actual homotopy groups of these spaces. Our proof relies on results about 2-divisibility of restricted sums of binomial coefficients times powers proved by the author in a companion paper.Comment: 8 page