It is proved that, if $P$ is a polynomial with integer coefficients, having degree 2, and $1>\varepsilon>0$, then $n(n-1)\cdots(n-k+1)=P(m)$ has only finitely many natural solutions $(m,n,k)$, $n\ge k>n\varepsilon$, provided that the $abc$ conjecture is assumed to hold under Szpiro's formulation.

A unital C*-algebra $A$ is said to have cancellation of projections if the semigroup $D(A)$ of Murray–von Neumann equivalence classes of projections in matrices over $A$ is cancellative. It has long been known that stable rank one implies cancellation for any $A$, and some partial converses have been established. In this paper it is proved that cancellation does not imply stable rank one for simple, stably finite C*-algebras.

Let $p{>}2$ be a prime, and let $k$ be a field of characteristic zero, linearly disjoint from the $p$th cyclotomic extension of $\Q$. Given a projective Galois representation $\varrho \colon \Gal(\kbar/k) {\To} \PGL_2(\F_p)$ with cyclotomic determinant, two twists $X_\varrho(p)$ and $ X'_\varrho(p)$ of a certain rational model of the modular curve $X(p)$ can be attached to it. The $k$-rational points of these twists classify the elliptic curves $E/k$ such that ${\rhobar}_{E, p}{=}\varrho$, where ${\rhobar}_{E, p}$ denotes the projective Galois representation associated with the $p$-torsion module $E[p]$. The octahedral ($p{=}3$) and icosahedral ($p{=}5$) genus-zero cases are discussed in further detail.

Crispin St John Alvah Nash-Williams was born in Cardiff on 19 December 1932. His father was Keeper of Archaeology at the National Museum of Wales and also Senior Lecturer in Archaeology at University College, Cardiff. War broke out when Crispin was six; his father joined the army and his consequent absence changed the family's life.