Abstract
These are notes on Zhang's work and subsequent developments produced in preparation for 5 hours of talks for a general mathematical audience given in Cambridge, Edinburgh and Auckland over the last year. Being for colloquium-style talks, these notes are at a much lower level than other accounts in the literature. In places they are deliberately quite nonrigorous. Whether anyone will find them helpful is unclear but some care was taken on their preparation and there is, I hope, little harm in making them available here.
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