In this paper, we look at numbers of the form
Hr,k:=Fk−1Fr−k+2+FkFr−k. These numbers are the entries of a
triangular array called the \emph{determinant Hosoya triangle} which we denote
by 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 H. Since the Fibonacci and Lucas numbers
appear as entries in H, our research is an extension of the
classical questions concerning whether there are infinitely many Fibonacci or
Lucas primes. We prove that H has arbitrarily large neighbourhoods
of composite entries. Finally we present an abundance of data indicating a very
high density of primes in H.Comment: two figure