Mathematics Lecture Series, Fall 2013

Fresno State

DR. STEFAAN DELCROIX

California State University, Fresno

“Bertrand’s Postulate”

Friday, December 6, 2013

from 3:00 p.m. to 4:00 p.m.

Science 2, room 308

In 1845, Joseph Bertrand conjectured what is now known as Bertrand’s Postulate:

For all n>1, there is at least one prime number between n and 2n.

Bertrand was not able to prove his conjecture. It was proved by Chebyshev (1850), Ramanujan (1919) and Erdős (1932) but their proofs used rather advanced techniques in number theory.

We will prove the following generalization:

Let k≥1. Then for any n ≥ max {4000,41 k^2}, there are at least k primes between n and 2n.

The proof we present here is elementary but extremely elegant and follows Erdős (1934). It is included in “Proofs from THE BOOK”. In this talk, we hope to convince students that plenty of topics in number theory are accessible to undergraduate and graduate students.

If you need a disability-related accommodation or wheelchair access information, please contact Carmen Caprau at (559) 278-4997 or e-mail ccaprau@csufresno.edu. Requests should be made at least one week in advance of the event.