Блок-схемы, группы перестановок и простые значения многочленов.
Блок-схемы, группы перестановок и простые значения многочленов.
A recent construction by Amarra, Devillers and Praeger of block designs with specific parameters and large symmetry groups depends on certain quadratic polynomials, with integer coefficients, taking prime power values. Similarly, a recent construction by Hujdurović, Kutnar, Kuzma, Marušič, Miklavič and Orel of permutation groups with specific intersection densities depends on certain cyclotomic polynomials taking prime values. The Bunyakovsky Conjecture, if true, would imply that each of these polynomials takes infinitely many prime values, giving infinite families of block designs and permutation groups with the required properties. We have found large numbers of prime values of these polynomials, and the numbers found agree very closely with the estimates for them provided by Li’s recent modification of the Bateman–Horn Conjecture. While this does not prove that these polynomials take infinitely many prime values, it provides strong evidence for this, and it also adds extra support for the validity of the Bunyakovsky and Bateman–Horn Conjectures.
Bateman–Horn Conjecture, Block design, Bunyakovsky Conjecture, intersection density, permutation group, polynomial, prime number
233-253
Jones, Gareth A.
fdb7f584-21c5-4fe4-9e57-b58c78ebe3f5
Zvonkin, Alexander K.
a3e21930-67d2-486f-90c3-0a4529352b35
Jones, Gareth A.
fdb7f584-21c5-4fe4-9e57-b58c78ebe3f5
Zvonkin, Alexander K.
a3e21930-67d2-486f-90c3-0a4529352b35
Jones, Gareth A. and Zvonkin, Alexander K.
(2023)
Блок-схемы, группы перестановок и простые значения многочленов.
Trudy Instituta Matematiki i Mekhaniki UrO RAN, 29 (1), .
(doi:10.21538/0134-4889-2023-29-1-233-253).
(In Press)
Abstract
A recent construction by Amarra, Devillers and Praeger of block designs with specific parameters and large symmetry groups depends on certain quadratic polynomials, with integer coefficients, taking prime power values. Similarly, a recent construction by Hujdurović, Kutnar, Kuzma, Marušič, Miklavič and Orel of permutation groups with specific intersection densities depends on certain cyclotomic polynomials taking prime values. The Bunyakovsky Conjecture, if true, would imply that each of these polynomials takes infinitely many prime values, giving infinite families of block designs and permutation groups with the required properties. We have found large numbers of prime values of these polynomials, and the numbers found agree very closely with the estimates for them provided by Li’s recent modification of the Bateman–Horn Conjecture. While this does not prove that these polynomials take infinitely many prime values, it provides strong evidence for this, and it also adds extra support for the validity of the Bunyakovsky and Bateman–Horn Conjectures.
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Accepted/In Press date: 4 June 2023
Additional Information:
Funding Information:
1This paper is based on the results of the 2021 Conference of International Mathematical Centers “Groups and Graphs, Semigroups and Synchronization”. 2Alexander Zvonkin was partially supported by the ANR project Combiné (ANR-19-CE48-0011).
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© 2023 Authors. All rights reserved.
Alternative titles:
Block Designs, Permutation Groups and Prime Values of Polynominals
Keywords:
Bateman–Horn Conjecture, Block design, Bunyakovsky Conjecture, intersection density, permutation group, polynomial, prime number
Identifiers
Local EPrints ID: 478568
URI: http://eprints.soton.ac.uk/id/eprint/478568
ISSN: 0134-4889
PURE UUID: 1b2ebbe2-af83-4421-aa15-eb6ce0ff237c
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Date deposited: 04 Jul 2023 18:11
Last modified: 05 Jun 2024 18:32
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Author:
Gareth A. Jones
Author:
Alexander K. Zvonkin
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