The Department of Mathematics is organising a Department Colloquium on the topic “Primes in the Interval ([kx,(k+1)x]) (Joint work with R. Higa and R. Sugawara)” by Prof. Hiroki Aoki, Associate Professor, Department of Mathematics, Faculty of Science and Technology, Tokyo University of Science on June 04, 2026.
Abstract
Bertrand’s postulate: “For any natural number $n$, there always exists a prime greater than $n$ and less than $2n$” was first conjectured in 1845 by Bertrand and proven by Chebyshev in 1852. Later, Ramanujan made the proof considerably simpler, and Erd\"{o}s gave a completely elementary and concise proof in 1932. Then, is a slightly shorter interval in this postulate still valid? Here we consider the following proposition $P(k)$ for a natural number $k$: $P(k)$: “For a sufficiently large natural number $n$, there always exists a prime between $kn$ and $(k+1)n$.” Here we say it Bertrand-type prime distribution theorem. In modern high-level mathematics, by using the Prime Number Theorem (PNT), we can show that $P(k)$ is true for any $k$ easily. In addition, we know an elementary proof of PNT, although it is complex and long. However, it is not at all clear whether a concise, or even completely elementary, proof exists for general $k$, such as the one given by Ramanujan and Erd\”{o}s for Bertrand’s postulate. Previous research had suggested that a completely elementary proof of $P(k)$ based on the method of Erd\”{o}s is impossible when $k \ge 5$ (Balliet, 2015). However, by adding some new idea, we find a completely elementary proof of $P(k)$ for $k \le 15 $ some new idea.
This colloquium provides an excellent opportunity to learn about recent advances in Number Theory and interact with an internationally renowned mathematician. All faculty members, research scholars, postdoc researchers, and students are cordially invited to attend and participate in the discussion.
