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A305048 Number of ordered pairs (k, m) of nonnegative integers such that 5^k + 10^m is not only a primitive root modulo prime(n) but also smaller than prime(n).

%I A305048 #12 May 25 2018 06:02:16
%S A305048 0,1,1,0,2,3,2,2,2,4,1,3,5,1,4,3,3,4,2,2,3,2,4,4,2,5,4,4,2,2,2,4,4,6,
%T A305048 6,4,3,3,7,6,6,2,4,3,5,3,2,3,8,3,4,4,1,3,5,5,6,5,6,4,3,5,1,1,3,4,4,2,
%U A305048 7,2,4,4,2,8,3,7,7,3,5,4,6,1,3,4,4,7,5,4,6,2
%N A305048 Number of ordered pairs (k, m) of nonnegative integers such that 5^k + 10^m is not only a primitive root modulo prime(n) but also smaller than prime(n).
%C A305048 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.
%C A305048 We have verified this for any prime p > 7 not exceeding 10^9.
%C A305048 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.
%C A305048 See also A305048 for similar conjectures.
%H A305048 Zhi-Wei Sun, <a href="/A305048/b305048.txt">Table of n, a(n) for n = 1..100000</a>
%H A305048 Zhi-Wei Sun, <a href="http://arxiv.org/abs/1405.0290">New observations on primitive roots modulo primes</a>, arXiv:1405.0290 [math.NT], 2014.
%H A305048 Zhi-Wei Sun, <a href="http://maths.nju.edu.cn/~zwsun/160p.pdf">Problems on combinatorial properties of primes</a>, 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.
%e A305048 a(14) = 1 with 5^2 + 10^0 = 26 a primitive root modulo prime(14) = 43.
%e A305048 a(101) = 1 with 5^0 + 10^0 = 2 a primitive root modulo prime(101) = 547.
%e A305048 a(111) = 1 with 5^2 + 10 = 35 a primitive root modulo prime(111) = 607.
%e A305048 a(5718) = 1 with 5^0 + 10^3 = 1001 a primitive root modulo prime(5718) = 56401.
%e A305048 a(6613) = 1 with 5^1 + 10^3 = 1005 a primitive root modulo prime(6613) = 66301.
%e A305048 a(430516) = 1 with 5^5 + 10^1 = 3135 a primitive root modulo prime(430516) = 6276271.
%t A305048 p[n_]:=p[n]=Prime[n];
%t A305048 Dv[n_]:=Dv[n]=Divisors[n];
%t A305048 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;
%t A305048 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]
%Y A305048 A000040, A000351, A011557, A303540, A239957, A241476, A241504, A241516, A305030.
%K A305048 nonn
%O A305048 1,5
%A A305048 _Zhi-Wei Sun_, May 24 2018