The Department of Mathematics successfully organised a Department Colloquium on “Primes in the Interval [kx,(k+1)x]” on June 04, 2026. The colloquium was delivered by Prof. Hiroki Aoki, Associate Professor, Department of Mathematics, Faculty of Science and Technology, Tokyo University of Science.

The lecture focused on the distribution of prime numbers in short intervals, a standard topic in analytic and elementary number theory. Prof. Aoki began by discussing Bertrand’s Postulate, which states that for every natural number n, there exists at least one prime number between n and 2n. He provided historical insights into its development, highlighting contributions by Bertrand, Chebyshev, Ramanujan, and Erdős.

The speaker then introduced a generalisation of Bertrand’s Postulate through the proposition P(k), which asserts that for sufficiently large natural numbers n, there exists a prime number between kn and (k+1)n. The discussion explored the significance of Bertrand-type prime distribution theorems and their relationship with the Prime Number Theorem.

A major highlight of the colloquium was the presentation of recent joint work with R. Higa and R. Sugawara, in which the authors established completely elementary proofs of P(k) for k ≤ 15. The speaker explained the motivation behind the work, previous limitations in the literature, and the novel ideas that enabled these results.

The session concluded with an engaging question-and-answer segment. Faculty members, research scholars, and students actively participated by raising questions about elementary methods in number theory, the limitations of existing techniques, and possible extensions of the results presented to larger values of k. Prof. Aoki provided detailed explanations and shared perspectives on future directions of research in this area.

The colloquium was highly informative and intellectually stimulating. It provided participants with a deeper understanding of prime distribution problems and contemporary research developments in number theory.