4 research outputs found
On Grosswald's conjecture on primitive roots
Grosswald's conjecture is that , the least primitive root modulo ,
satisfies for all . We make progress towards
this conjecture by proving that for all and for all .Comment: 7 page
Permuting operations on strings and the distribution of their prime numbers
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 {Xn}nā„2} of similar permutations. We call an integer n X-prime if Xn consists of a single cycle of length n(nā„2). 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
Artin's primitive root conjecture -a survey -
This is an expanded version of a write-up of a talk given in the fall of 2000
in Oberwolfach. A large part of it is intended to be understandable by
non-number theorists with a mathematical background. The talk covered some of
the history, results and ideas connected with Artin's celebrated primitive root
conjecture dating from 1927. In the update several new results established
after 2000 are also discussed.Comment: 87 pages, 512 references, to appear in Integer