# 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

### 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