Primitive roots for Pjateckii-Šapiro primes

Abstract

For any non-integral positive real number c, any sequence (bnc)n is called a Pjateckii-Šapiro sequence. Given a real number c in the interval ( 1, 12 11 ) , it is known that the number of primes in this sequence up to x has an asymptotic formula. We would like to use the techniques of Gupta and Murty to study Artin’s problems for such primes. We will prove that even though the set of Pjateckii-Šapiro primes is of density zero for a fixed c, one can show that there exist natural numbers which are primitive roots for infinitely many Pjateckii-Šapiro primes for any fixed c in the interval ( 1, √ 77 7 − 1 4 ) .