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On the Average of the Least Primitive Root Modulo p
Author(s) -
Elliott P. D. T. A.,
Murata Leo
Publication year - 1997
Publication title -
journal of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.441
H-Index - 62
eISSN - 1469-7750
pISSN - 0024-6107
DOI - 10.1112/s0024610797005310
Subject(s) - primitive root modulo n , modulo , mathematics , prime (order theory) , combinatorics , root (linguistics) , moduli , constant (computer programming) , prime number , discrete mathematics , arithmetic , physics , computer science , quantum mechanics , philosophy , linguistics , programming language
In this paper we study the value distribution of the least primitive root to a prime modulus, as the modulus varies. For each odd prime number p , we shall denote by g ( p ) and G ( p ) the least primitive root and the least prime primitive root (mod p ), respectively. Numerical examples show that, in most cases, g ( p ) and G ( p ) are very small (cf. §4). We can support this observation by a probabilistic argument [ 14 , §1]. In fact, on the assumption of a good distribution of the primitive residue classes modulo p , we can surmise thatlim x → ∞ π( x )‐ 1∑ p ⩽ xg ( p ) = E with a positive constant E , ( 1.1 )where π( x ) denotes the number of primes not exceeding x .