Graduate Number Theory Seminar

Title: GCDs of Fibonacci numbers and their indices

Abstract: Questions about divisibility in recurrence sequences have a long history, ranging from classical work on Fibonacci and Lucas sequences to modern results on common divisors of exponentially growing sequences. We will begin with some of this history before focusing on gcd(n,F_n), where F_n denotes the n-th Fibonacci number. Which integers can occur as these GCDs? How frequently does a given value occur, and how large are they on average? We will conclude with recent work on moments of (\gcd(n,u_n)) for general Lucas sequences, jointly with Ayan Nath and Emanuele Tron. No prior knowledge of Lucas sequences will be assumed.