Permuting operations on strings and their relation to prime numbers

Some length-preserving operations on strings only permute the symbol positions in strings; such an operation XX gives rise to a family {Xn}n≥2{Xn}n≥2 of similar permutations. We investigate the structure and the order of the cyclic group generated by XnXn. We call an integer nn XX-prime   if XnXn consists of a single cycle of length nn (n≥2n≥2). Then we show some properties of these XX-primes, particularly, how XX-primes are related to X′X′-primes as well as to ordinary prime numbers. Here XX and X′X′ range over well-known examples (reversal, cyclic shift, shuffle, twist) and some new ones based on the Archimedes spiral and on the Josephus problem.

► Provides a complete characterization of twist primes (Queneau numbers) and its history.
► Achieved similar results for shuffle primes and Josephus(2) primes.
► Archimedes spirals and their primes result in a unifying approach.
► Characterizations are given in terms of finite fields of prime order.
► Includes a solution of the Josephus(2) problem and its dual problem.