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arXiv · 2211.10788

Primes and composites in the determinant Hosoya triangle

Hsin-Yun Ching·Rigoberto Flórez·F. Luca·Antara Mukherjee·J. C. Saunders
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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 ${\mathcal H}$.

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Hsin-Yun Ching, Rigoberto Flórez, F. Luca, Antara Mukherjee, J. C. Saunders. 2022-11-19. Primes and composites in the determinant Hosoya triangle. https://arxiv.org/abs/2211.10788

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