On some geometry of Mersenne primes
type: Item
instance of: scholarly article
title: On some geometry of Mersenne primes
zbMATH Open document ID: 0818.11006
DOI: 10.1007/BF01876871
author: Yeong-Nan Yeh
author: James R. Clay
published in: Periodica Mathematica Hungarica
publication date: 1995-08-24T00:00:00Z
review text: A possible connection between Mersenne primes and certain geometrical structures is implied. Here the authors consider the structures \((\mathbb{Z}_ q,{\mathcal B}_ p^ \#, \in)\) resulting from a planar nearring \((\mathbb{Z}_ q, +, *)\), where \(q= M_ p\) is a Mersenne prime, \(\mathbb{Z}_ q\) denotes the integers modulo \(q\), \(*\) is a multiplication of the Ferrero planar nearring factory manufactured by the subgroup of \(\mathbb{Z}_ q\) of order \(p\), \({\mathcal B}_ p^ \#\) is the set of circles \(C(r:b)= \mathbb{Z}_ p^* r+b\), \(r,b\in \mathbb{Z}_ q\), \(r\neq 0\), and \(\in\) is the usual element relation. It is shown that \((\mathbb{Z}_ q,{\mathcal B}_ p^ \#, \in)\) is circular, i.e. three distinct points of \(\mathbb{Z}_ q\) belong to at most one circle of \({\mathcal B}_ p^ \#\), and satisfies various properties which are not valid in the Euclidean plane and which have not been observed elsewhere. In particular, if \(k\) is the number of points on a circle \(C(r:0)\), then there are a maximum number, namely \(k-1\), of classes of circles \(E^ r_ b= \{C(r: b')\mid b'\in C(r:0)\}\) whose circles are tangent to \(C(r:0)\).
Mathematics Subject Classification ID: 11A41
Mathematics Subject Classification ID: 51A99
zbMATH DE Number: 736861
zbMATH Keywords: Mersenne primes
zbMATH Keywords: planar nearring
zbMATH Keywords: Ferrero planar nearring factory
zbMATH Keywords: classes of circles
reviewed by: W. Kerby
MaRDI profile type: Publication
cites work: Q5732661
cites work: Q4126346
cites work: Factorizations of 𝑏ⁿ±1, 𝑏=2, 3, 5, 6, 7, 10, 11, 12 Up to High Powers
cites work: Applications of planar near-rings to geometry and combinatorics
cites work: Q4733869
cites work: Q3819616
cites work: Tactical configurations from a planar nearring can also generate balanced incomplete block designs
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description: scientific article; zbMATH DE number 736861