PRIMES AND COMPOSITES IN THE DETERMINANT HOSOYA TRIANGLE

Hsin Yun Ching; Rigoberto Flórez; F. Luca; Antara Mukherjee; J. C. Saunders

doi:10.1080/00150517.2022.12427433

PRIMES AND COMPOSITES IN THE DETERMINANT HOSOYA TRIANGLE

Hsin Yun Ching, Rigoberto Flórez, F. Luca, Antara Mukherjee, J. C. Saunders

Research output: Contribution to journal › Article › peer-review

Abstract

In this paper, we look at numbers of the form H_r,k := F_k−₁F_r−k₊₂ + F_kF_r−k. These numbers are the entries of a triangular array called the 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.

Original language	English
Pages (from-to)	56-110
Number of pages	55
Journal	Fibonacci Quarterly
Volume	60
Issue number	5
DOIs	https://doi.org/10.1080/00150517.2022.12427433
State	Published - 1 Dec 2022
Externally published	Yes

ASJC Scopus subject areas

Algebra and Number Theory

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title = "PRIMES AND COMPOSITES IN THE DETERMINANT HOSOYA TRIANGLE",

abstract = "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 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.",

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AU - Ching, Hsin Yun

AU - Flórez, Rigoberto

AU - Luca, F.

AU - Mukherjee, Antara

AU - Saunders, J. C.

PY - 2022/12/1

Y1 - 2022/12/1

N2 - 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 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.

AB - 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 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.

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JO - Fibonacci Quarterly

JF - Fibonacci Quarterly

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ER -

PRIMES AND COMPOSITES IN THE DETERMINANT HOSOYA TRIANGLE

Abstract

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