Conformally invariant powers of the Laplacian -- A complete non-existence theorem

We show that on conformal manifolds of even dimension $n\ge 4$ there is no conformally invariant natural differential operator between density bundles with leading part a power of the Laplacian ${\mathrm{\Delta }}^k$ for $k>n\mathrm{/}2$. This shows that a large class of invariant operators on conformally flat manifolds do not generalise to arbitrarily curved manifolds and that the theorem of Graham, Jenne, Mason and Sparling, asserting the existence of curved version of ${\mathrm{\Delta }}^k$ for $1\le k\le n\mathrm{/}2$, is sharp.