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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 atriangular array called the \emph{determinant Hosoya triangle} which we denoteby ${\mathcal H}$. We discuss the divisibility properties of the above numbersand their primality. We give a small sieve of primes to illustrate the densityof prime numbers in ${\mathcal H}$. Since the Fibonacci and Lucas numbersappear as entries in ${\mathcal H}$, our research is an extension of theclassical questions concerning whether there are infinitely many Fibonacci orLucas primes. We prove that ${\mathcal H}$ has arbitrarily large neighbourhoodsof composite entries. Finally we present an abundance of data indicating a veryhigh density of primes in ${\mathcal H}$.

Primes and composites in the determinant Hosoya triangle