An incomplete variant of Wilson's congruence

This article examines the nontrivial solutions of the congruence $$(p-1)\cdots(p-r) \equiv -1 \pmod p.$$ We discuss heuristics for the proportion of primes $p$ that have exactly $N$ solutions to this congruence. We supply numerical evidence in favour of these conjectures, and discuss the algorithms used in our calculations.Comment: 7 pages, 2 table

Zeroes of partial sums of the zeta-function

This article considers the positive integers $N$ for which $\zeta_{N}(s) = \sum_{n=1}^{N} n^{-s}$ has zeroes in the half-plane $\Re(s)>1$. Building on earlier results, we show that there are no zeroes for $1\leq N\leq 18$ and for $N=20, 21, 28$. For all other $N$ there are infinitely many zeroes.Comment: 5 Pages - Final Version will appear in LMS JC

On the Sum of Two Squares and At Most Two Powers of 2

We demonstrate that there are infinitely many integers that cannot be expressed as the sum of two squares of integers and up to two non-negative integer powers of 2.Comment: 5 pages; to appear in Amer. Math. Monthl

Optimal translational termination requires C4 lysyl hydroxylation of eRF1

Efficient stop codon recognition and peptidyl-tRNA hydrolysis are essential in order to terminate translational elongation and maintain protein sequence fidelity. Eukaryotic translational termination is mediated by a release factor complex that includes eukaryotic release factor 1 (eRF1) and eRF3. The N terminus of eRF1 contains highly conserved sequence motifs that couple stop codon recognition at the ribosomal A site to peptidyl-tRNA hydrolysis. We reveal that Jumonji domain-containing 4 (Jmjd4), a 2-oxoglutarate- and Fe(II)-dependent oxygenase, catalyzes carbon 4 (C4) lysyl hydroxylation of eRF1. This posttranslational modification takes place at an invariant lysine within the eRF1 NIKS motif and is required for optimal translational termination efficiency. These findings further highlight the role of 2-oxoglutarate/Fe(II) oxygenases in fundamental cellular processes and provide additional evidence that ensuring fidelity of protein translation is a major role of hydroxylation