  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 ${\mathcal H}$.
