Permuting Operations on Strings and the Distribution of Their Prime Numbers

P.R.J. Asveld

Research output: Contribution to journalArticleAcademicpeer-review

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Abstract

Several ways of interleaving, as studied in theoretical computer science, and some subjects from mathematics can be modeled by length-preserving operations on strings, that only permute the symbol positions in strings. Each such operation $X$ gives rise to a family $\{X_n\}_{n\geq2}$ of similar permutations. We call an integer $n$ {\em $X$-prime} if $X_n$ consists of a single cycle of length $n$ ($n\geq2$). For some instances of $X$ -- such as shuffle, twist, operations based on the Archimedes'spiral and on the Josephus problem -- we investigate the distribution of $X$-primes and of the associated (ordinary) prime numbers, which leads to variations of some well-known conjectures on the density of certain sets of prime numbers.
Original languageEnglish
Pages (from-to)1868-1881
Number of pages14
JournalDiscrete applied mathematics
Volume161
Issue number13-14
DOIs
Publication statusPublished - Sep 2013

Fingerprint

Distribution of Primes
Prime number
Computer science
Strings
Josephus problem
Spiral of Archimedes
Shuffle
Interleaving
Twist
Permutation
Computer Science
Cycle
Integer

Keywords

  • HMI-SLT: Speech and Language Technology
  • MSC-11A07
  • MSC-11N05
  • MSC-12E20
  • distribution of prime numbers
  • Artin's conjecture (on primitive roots)
  • Archimedes' spiral
  • Twist
  • Queneau number
  • Shuffle
  • Josephus problem
  • METIS-297691
  • MSC-68R15

Cite this

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title = "Permuting Operations on Strings and the Distribution of Their Prime Numbers",
abstract = "Several ways of interleaving, as studied in theoretical computer science, and some subjects from mathematics can be modeled by length-preserving operations on strings, that only permute the symbol positions in strings. Each such operation $X$ gives rise to a family $\{X_n\}_{n\geq2}$ of similar permutations. We call an integer $n$ {\em $X$-prime} if $X_n$ consists of a single cycle of length $n$ ($n\geq2$). For some instances of $X$ -- such as shuffle, twist, operations based on the Archimedes'spiral and on the Josephus problem -- we investigate the distribution of $X$-primes and of the associated (ordinary) prime numbers, which leads to variations of some well-known conjectures on the density of certain sets of prime numbers.",
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author = "P.R.J. Asveld",
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journal = "Discrete applied mathematics",
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Permuting Operations on Strings and the Distribution of Their Prime Numbers. / Asveld, P.R.J.

In: Discrete applied mathematics, Vol. 161, No. 13-14, 09.2013, p. 1868-1881.

Research output: Contribution to journalArticleAcademicpeer-review

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N2 - Several ways of interleaving, as studied in theoretical computer science, and some subjects from mathematics can be modeled by length-preserving operations on strings, that only permute the symbol positions in strings. Each such operation $X$ gives rise to a family $\{X_n\}_{n\geq2}$ of similar permutations. We call an integer $n$ {\em $X$-prime} if $X_n$ consists of a single cycle of length $n$ ($n\geq2$). For some instances of $X$ -- such as shuffle, twist, operations based on the Archimedes'spiral and on the Josephus problem -- we investigate the distribution of $X$-primes and of the associated (ordinary) prime numbers, which leads to variations of some well-known conjectures on the density of certain sets of prime numbers.

AB - Several ways of interleaving, as studied in theoretical computer science, and some subjects from mathematics can be modeled by length-preserving operations on strings, that only permute the symbol positions in strings. Each such operation $X$ gives rise to a family $\{X_n\}_{n\geq2}$ of similar permutations. We call an integer $n$ {\em $X$-prime} if $X_n$ consists of a single cycle of length $n$ ($n\geq2$). For some instances of $X$ -- such as shuffle, twist, operations based on the Archimedes'spiral and on the Josephus problem -- we investigate the distribution of $X$-primes and of the associated (ordinary) prime numbers, which leads to variations of some well-known conjectures on the density of certain sets of prime numbers.

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KW - MSC-12E20

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KW - Artin's conjecture (on primitive roots)

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KW - Queneau number

KW - Shuffle

KW - Josephus problem

KW - METIS-297691

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