# Permuting Operations on Strings and the Distribution of Their Prime Numbers

P.R.J. Asveld

Research output: Book/ReportReportProfessional

### 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 $n$ $X$-{\em 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 in number theory.
Original language Undefined Enschede Centre for Telematics and Information Technology (CTIT) 22 Published - 17 Oct 2011

### Publication series

Name CTIT Technical Report Series University of Twente, Centre for Telematics and Information Technology TR-CTIT-11-24 1381-3625

### Keywords

• MSC-68R15
• EWI-20685
• distribution of prime numbers
• Shuffle
• Josephus problem
• Queneau number
• Twist
• Artin's conjecture (on primitive roots)
• Archimedes' spiral
• IR-78281
• METIS-278873
• MSC-11B25
• MSC-11A41
• MSC-11A07
• HMI-SLT: Speech and Language Technology

### Cite this

Asveld, P. R. J. (2011). Permuting Operations on Strings and the Distribution of Their Prime Numbers. (CTIT Technical Report Series; No. TR-CTIT-11-24). Enschede: Centre for Telematics and Information Technology (CTIT).
Asveld, P.R.J. / Permuting Operations on Strings and the Distribution of Their Prime Numbers. Enschede : Centre for Telematics and Information Technology (CTIT), 2011. 22 p. (CTIT Technical Report Series; TR-CTIT-11-24).
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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 $n$ $X$-{\em 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 in number theory.",
keywords = "MSC-68R15, EWI-20685, distribution of prime numbers, Shuffle, Josephus problem, Queneau number, Twist, Artin's conjecture (on primitive roots), Archimedes' spiral, IR-78281, METIS-278873, MSC-11B25, MSC-11A41, MSC-11A07, HMI-SLT: Speech and Language Technology",
author = "P.R.J. Asveld",
note = "eemcs-eprint-20685",
year = "2011",
month = "10",
day = "17",
language = "Undefined",
series = "CTIT Technical Report Series",
publisher = "Centre for Telematics and Information Technology (CTIT)",
number = "TR-CTIT-11-24",

}

Asveld, PRJ 2011, Permuting Operations on Strings and the Distribution of Their Prime Numbers. CTIT Technical Report Series, no. TR-CTIT-11-24, Centre for Telematics and Information Technology (CTIT), Enschede.
Enschede : Centre for Telematics and Information Technology (CTIT), 2011. 22 p. (CTIT Technical Report Series; No. TR-CTIT-11-24).

Research output: Book/ReportReportProfessional

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T1 - Permuting Operations on Strings and the Distribution of Their Prime Numbers

AU - Asveld, P.R.J.

N1 - eemcs-eprint-20685

PY - 2011/10/17

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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 $n$ $X$-{\em 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 in number theory.

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 $n$ $X$-{\em 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 in number theory.

KW - MSC-68R15

KW - EWI-20685

KW - distribution of prime numbers

KW - Shuffle

KW - Josephus problem

KW - Queneau number

KW - Twist

KW - Artin's conjecture (on primitive roots)

KW - Archimedes' spiral

KW - IR-78281

KW - METIS-278873

KW - MSC-11B25

KW - MSC-11A41

KW - MSC-11A07

KW - HMI-SLT: Speech and Language Technology

M3 - Report

T3 - CTIT Technical Report Series

BT - Permuting Operations on Strings and the Distribution of Their Prime Numbers

PB - Centre for Telematics and Information Technology (CTIT)

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ER -

Asveld PRJ. Permuting Operations on Strings and the Distribution of Their Prime Numbers. Enschede: Centre for Telematics and Information Technology (CTIT), 2011. 22 p. (CTIT Technical Report Series; TR-CTIT-11-24).