To begin, I will describe a new result on the greatest common divisor of binomial coefficients \({n \choose q}, {n \choose 2q}, {n \choose 3q},\dots\) A result related to the geometry of manifolds, answering a question raised the last time I spoke in this seminar, and recently published in the Amer. Math. Monthly.

Next, I will describe a new proof that \(\mathrm{tmf}[1/6]\) is not a ring spectrum quotient of \(\mathrm{MO}\langle8\rangle[1/6]\). In fact, for any prime \(p>3\) and any sequence \(X\) of homogeneous elements of \(\pi_*\mathrm{MO}\langle8\rangle\), the \(\pi_*\mathrm{MO}\langle8\rangle\)-module

\[\pi_*\big(\mathrm{MO}\langle8\rangle_{(p)}/X\big)\]

is not (even abstractly) isomorphic to \(\pi_*\mathrm{tmf}_{(p)}\). The key is showing that, for any commutative ring spectrum \(R\) and any sequence \(X\) of homogeneous elements of \(\pi_*(R)\), there is an isomorphism of graded \(\mathbf{Q}\)-vector spaces

\[\pi_*(R/X)\otimes\mathbf{Q} \cong \mathrm{H}_*(\mathrm{Tot}(\mathrm{K}(X)))\otimes\mathbf{Q},\]

where the right-hand side is the rational homology of the (total) Koszul complex of \(X\), which is strictly bigger than \(\pi_*(R)/(X)\otimes\mathbf{Q}\) unless \(X\) is a \(\pi_*(R)\otimes\mathbf{Q}\)-quasi-regular sequence. The result then follows from the fact that the kernel of the \(p\)-local Witten genus cannot be generated by a \(\pi_*\mathrm{MO}\langle8\rangle\otimes\mathbf{Q}\)-quasi-regular sequence.