Mathematics Colloquium: Exploring Properties of Twin Primes and Brun’s Theorem by Yuan Qiu ’24, Monday February 19, 1:00 – 1:50pm, North Science Building 113, Wachenheim
Abstract: We explore properties of twin primes, which are primes differing by two. We first discuss infinity of primes and distribution of primes, leading to the natural (and still unsolved) question on whether or not there are infinitely many twin primes. We then present Brun’s Theorem, which states that the sum of the reciprocals of the twin primes converges; as the sum of the reciprocals of the primes diverges, this implies that even if there are infinitely many twin primes, their density is much less than that of the primes. We show that Brun’s Theorem is a consequence of determining a sufficiently good upper bound on the number of twin primes at most x, and sketch the inclusion-exclusion argument behind the proof.