Number Theory Seminar

J.C. Saunders, Middle Tennessee State University

"Primes and composites in the determinant Hosoya triangle"

In this talk, we look at numbers of the form $H_{r,k} := F_{k-1}F_{r-k+2} +F_kF_{r-k},$ where $F_n$ denotes the $n$th Fibonacci number. These numbers are the entries of a triangular array called the {\it determinant Hosoya triangle}\/ which we denote by $\mathcal{H}.$ We discuss the divisibility properties of the above numbers and their primality. We give a small sieve to illustrate the density of prime numbers in $\mathcal{H}.$ Since the Fibonacci and Lucas numbers appear as entries in $\mathcal{H},$ this research is an extension of the classical questions concerning whether there are infinitely many Fibonacci or Lucas primes. We prove that $\mathcal{H}$ has arbitrarily large neighbourhoods of composite entries. Finally we present an abundance of data indicating a very high density of primes in $\mathcal{H}.$ This is joint work with Hsin-Yun Ching, Rigoberto Fl\'orez, Florian Luca, and Antara Mukherjee.

MC 5417