Complete Resolution of the Polynomial Analogue of the Brocard–Ramanujan Problem in Mathematics

A Breakthrough 150 Years After the Problem Was First Proposed

Dr. Wataru Takeda, Department of Mathematics at the Faculty of Science, Toho University, has completely solved the polynomial analogue of the long-standing Brocard–Ramanujan problem, originally proposed in 1876 and still unsolved in its integer form.
This research has been published online in Finite Fields and Their Applications on September 30, 2025, and formally released in February 2026.

Figure: Integers and polynomials are known to exhibit similar structural properties.The polynomial analogue of the Brocard–Ramanujan problem—previously unsolved for integers—is presented here, along with its complete resolution.

Key Points

Complete solution of the polynomial analogue of the Brocard–Ramanujan problem, originally proposed in 1876 by the French mathematician Henri Brocard.
The original integer version of the Brocard–Ramanujan problem remains unsolved, and this breakthrough in the polynomial setting is expected to provide key insights toward solving the classical problem.

Journal:
Finite Fields and Their Applications (Volume 110, February 2026; online publication: September 30, 2025)
Title:
Brocard–Ramanujan problem for polynomials over finite fields
Author:
Wataru Takeda*
DOI:
10.1016/j.ffa.2025.102731

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