TY - JOUR
T1 - Uniform Distribution of Fractional Parts Related to Pseudoprimes
AU - Banks, William D.
AU - Garaev, Moubariz Z.
AU - Luca, Florian
AU - Shparlinski, Igor E.
PY - 2009/6
Y1 - 2009/6
N2 - We estimate exponential sums with the Fermat-like quotients f g(n)=gn-1-1/n and hg=gn-1 -1/p(n), where g and n are positive integers, n is composite, and P(n) is the largest prime factor of n. Clearly, both fg(n) and hg(n) are integers if n is a Fermat pseudoprime to base g, and if n is a Carmichael number, this is true for all g coprime to n. Nevertheless, our bounds imply that the fractional parts {fg(n)} and {hg(n)} are uniformly distributed, on average over g for fg(n), and individually for h g(n). We also obtain similar results with the functions &flm;g(n) = gfg(n) and hg(n) - gh g(n).
AB - We estimate exponential sums with the Fermat-like quotients f g(n)=gn-1-1/n and hg=gn-1 -1/p(n), where g and n are positive integers, n is composite, and P(n) is the largest prime factor of n. Clearly, both fg(n) and hg(n) are integers if n is a Fermat pseudoprime to base g, and if n is a Carmichael number, this is true for all g coprime to n. Nevertheless, our bounds imply that the fractional parts {fg(n)} and {hg(n)} are uniformly distributed, on average over g for fg(n), and individually for h g(n). We also obtain similar results with the functions &flm;g(n) = gfg(n) and hg(n) - gh g(n).
UR - http://www.scopus.com/inward/record.url?scp=67649985871&partnerID=8YFLogxK
U2 - 10.4153/CJM-2009-025-2
DO - 10.4153/CJM-2009-025-2
M3 - Article
AN - SCOPUS:67649985871
SN - 0008-414X
VL - 61
SP - 481
EP - 502
JO - Canadian Journal of Mathematics
JF - Canadian Journal of Mathematics
IS - 3
ER -