# Mike Roth (Queen's University)

Date

Tuesday September 26, 20234:00 pm - 5:00 pm

Location

Jeffery Hall, Room 319## Number Theory Seminar

**Tuesday, September 26th, 2023**

**Time:** 4:00 p.m. **Place:** Jeffery Hall, Room 319

**Speaker:** Mike Roth (Queen's University)

**Title:** The Bugeaud-Corvaja-Zannier theorem and extensions.

**Abstract:** Two integers a and b are called multiplicatively independent if the only solution (m,n) to a^m = b^n is (m,n)=(0,0). In 2003 Bugeaud, Corvaja, and Zannier proved that, if a and b are multiplicatively independent, then for every epsilon > 0 the inequality log gcd(a^n -1, b^n -1) < (epsilon) n holds for all but finitely many n > 0. In this talk we will discuss the BCZ result and give a new proof of their theorem. This proof makes it easier to see the key ingredients which make the argument work. The method of proof applies equally well to prove an extension of the theorem due to Corvaja and Zannier, and a more general extension due to Aaron Levin.

This is joint work with David McKinnon at Waterloo.