Primes and composites in the determinant Hosoya triangle

TL;DR

This paper investigates the divisibility and primality properties of numbers in the determinant Hosoya triangle, revealing a high density of primes and the presence of large composite neighborhoods, extending classical Fibonacci prime questions.

Contribution

It introduces the determinant Hosoya triangle, analyzes its divisibility and primality properties, and provides empirical data on prime density within this new structure.

Findings

High density of primes in the triangle

Existence of arbitrarily large composite neighborhoods

Extension of Fibonacci and Lucas prime questions

Abstract

In this paper, we look at numbers of the form $H_{r,k}:=F_{k-1}F_{r-k+2}+F_{k}F_{r-k}$. These numbers are the entries of a triangular array called the \emph{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 of primes to illustrate the density of prime numbers in ${\mathcal H}$. Since the Fibonacci and Lucas numbers appear as entries in ${\mathcal H}$, our 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 ${\mathcal H}$.