# Manchester Number Theory Seminar - Hung Bui

Dates: | 7 March 2023 |
---|---|

Times: | 15:00 - 15:00 |

What is it: | Seminar |

Organiser: | Department of Mathematics |

Who is it for: | University staff, External researchers, Current University students |

Speaker: Hung Bui (University of Manchester)

Title: Brocard-Ramanujan's problem on polynomial-factorial equations

Abstract: Brocard and Ramanujan, independently, posed the problem of finding all integer solutions to the equation n! + 1 = x^2. It is conjectured that the only solutions are n = 4,5,7. More generally, one may ask about integer solutions to the polynomial-factorial equation n! = P(x), where P is an integer polynomial of degree at least 2. Berend and Osgood showed that the set of n for which the equation has an integer solution x is of density 0, which answered a question of Erdos. We improve on this result of Berend and Osgood, obtaining a power saving bound for the number of solutions of a polynomial-factorial equation. This is joint work with Kyle Pratt and Alexandru Zaharescu.

Room: Frank Adams 1