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

P.R.J. Asveld

### 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 language English 1868-1881 14 Discrete applied mathematics 161 13-14 https://doi.org/10.1016/j.dam.2013.03.002 Published - 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.",
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",
author = "P.R.J. Asveld",
note = "eemcs-eprint-23421",
year = "2013",
month = "9",
doi = "10.1016/j.dam.2013.03.002",
language = "English",
volume = "161",
pages = "1868--1881",
journal = "Discrete applied mathematics",
issn = "0166-218X",
publisher = "Elsevier",
number = "13-14",

}

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

TY - JOUR

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

AU - Asveld, P.R.J.

N1 - eemcs-eprint-23421

PY - 2013/9

Y1 - 2013/9

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.

KW - HMI-SLT: Speech and Language Technology

KW - MSC-11A07

KW - MSC-11N05

KW - MSC-12E20

KW - distribution of prime numbers

KW - Artin's conjecture (on primitive roots)

KW - Archimedes' spiral

KW - Twist

KW - Queneau number

KW - Shuffle

KW - Josephus problem

KW - METIS-297691

KW - MSC-68R15

U2 - 10.1016/j.dam.2013.03.002

DO - 10.1016/j.dam.2013.03.002

M3 - Article

VL - 161

SP - 1868

EP - 1881

JO - Discrete applied mathematics

JF - Discrete applied mathematics

SN - 0166-218X

IS - 13-14

ER -