OFFSET
1,5
COMMENTS
Conjecture: a(n) > 0 for all n > 4. In other words, any prime p > 7 has a primitive root g < p of the form 5^k + 10^m with k and m nonnegative integers.
We have verified this for any prime p > 7 not exceeding 10^9.
It seems that a(n) = 1 only for n = 2, 3, 11, 14, 53, 63, 64, 82, 99, 101, 111, 129, 344, 369, 391, 795, 1170, 1587, 5629, 5718, 6613, 430516.
See also A305048 for similar conjectures.
LINKS
Zhi-Wei Sun, Table of n, a(n) for n = 1..100000
Zhi-Wei Sun, New observations on primitive roots modulo primes, arXiv:1405.0290 [math.NT], 2014.
Zhi-Wei Sun, Problems on combinatorial properties of primes, in: M. Kaneko, S. Kanemitsu and J. Liu (eds.), Number Theory: Plowing and Starring through High Wave Forms, Proc. 7th China-Japan Seminar (Fukuoka, Oct. 28--Nov. 1, 2013), Ser. Number Theory Appl., Vol. 11, World Sci., Singapore, 2015, pp. 169-187.
EXAMPLE
a(14) = 1 with 5^2 + 10^0 = 26 a primitive root modulo prime(14) = 43.
a(101) = 1 with 5^0 + 10^0 = 2 a primitive root modulo prime(101) = 547.
a(111) = 1 with 5^2 + 10 = 35 a primitive root modulo prime(111) = 607.
a(5718) = 1 with 5^0 + 10^3 = 1001 a primitive root modulo prime(5718) = 56401.
a(6613) = 1 with 5^1 + 10^3 = 1005 a primitive root modulo prime(6613) = 66301.
a(430516) = 1 with 5^5 + 10^1 = 3135 a primitive root modulo prime(430516) = 6276271.
MATHEMATICA
p[n_]:=p[n]=Prime[n];
Dv[n_]:=Dv[n]=Divisors[n];
gp[g_, p_]:=gp[g, p]=Mod[g, p]>0&&Sum[Boole[PowerMod[g, Dv[p-1][[k]], p]==1], {k, 1, Length[Dv[p-1]]-1}]==0;
tab={}; Do[r=0; Do[If[gp[5^a+10^b, p[n]], r=r+1], {a, 0, Log[5, p[n]-1]}, {b, 0, Log[10, p[n]-5^a]}]; tab=Append[tab, r], {n, 1, 90}]; Print[tab]
CROSSREFS
KEYWORD
nonn
AUTHOR
Zhi-Wei Sun, May 24 2018
STATUS
approved