TY - JOUR
T1 - On gaps between primitive roots in the hamming metric
AU - Dietmann, Rainer
AU - Elsholtz, Christian
AU - Shparlinski, Igor E.
PY - 2013/12
Y1 - 2013/12
N2 - We consider a modification of the classical number theoretic question about the gaps between consecutive primitive roots modulo a prime p, which by the well-known result of Burgess are known to be at most p1/4+o(1). Here we measure the distance in the Hamming metric and show that if p is a sufficiently large r-bit prime, then, for any integer n∈[1, p], one can obtain a primitive root modulo p by changing at most 0.11002786... r binary digits of n. This is stronger than what can be deduced from the Burgess result. Experimentally, the number of necessary bit changes is very small. We also show that each Hilbert cube contained in the complement of the primitive roots modulo p has dimension at most O(p1/5+ε), improving on previous results of this kind.
AB - We consider a modification of the classical number theoretic question about the gaps between consecutive primitive roots modulo a prime p, which by the well-known result of Burgess are known to be at most p1/4+o(1). Here we measure the distance in the Hamming metric and show that if p is a sufficiently large r-bit prime, then, for any integer n∈[1, p], one can obtain a primitive root modulo p by changing at most 0.11002786... r binary digits of n. This is stronger than what can be deduced from the Burgess result. Experimentally, the number of necessary bit changes is very small. We also show that each Hilbert cube contained in the complement of the primitive roots modulo p has dimension at most O(p1/5+ε), improving on previous results of this kind.
UR - http://www.scopus.com/inward/record.url?scp=84890601108&partnerID=8YFLogxK
U2 - 10.1093/qmath/has022
DO - 10.1093/qmath/has022
M3 - Article
AN - SCOPUS:84890601108
SN - 0033-5606
VL - 64
SP - 1043
EP - 1055
JO - Quarterly Journal of Mathematics
JF - Quarterly Journal of Mathematics
IS - 4
ER -