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A260947 Least positive integer k < prime(n) such that k = i + j for some 0 < i < j for which i,j,k and i*j*k are all primitive roots modulo prime(n), or 0 if no such k exists.

%I A260947 #9 Aug 15 2015 06:32:54
%S A260947 0,0,0,0,8,0,10,13,15,10,24,15,13,29,15,8,8,17,13,28,20,35,8,19,15,15,
%T A260947 11,7,24,17,29,8,24,15,10,13,20,18,15,5,8,28,47,15,5,41,29,11,8,31,17,
%U A260947 21,51,24,10,15,10,21,11,15
%N A260947 Least positive integer k < prime(n) such that k = i + j for some 0 < i < j for which i,j,k and i*j*k are all primitive roots modulo prime(n), or 0 if no such k exists.
%C A260947 Conjecture: a(n) > 0 for all n > 6. In other words, for any prime p > 13, there are three distinct elements a,b,c of {1,...,p-1} with a+b = c such that a,b,c and a*b*c are all primitive roots modulo p.
%H A260947 Zhi-Wei Sun, <a href="/A260947/b260947.txt">Table of n, a(n) for n = 1..10000</a>
%e A260947 a(5) = 8 since 2 + 6 = 8, and the four numbers 2, 6, 8 and 2*6*8=96 are all primitive roots modulo prime(5) = 11.
%t A260947 SQ[n_]:=IntegerQ[Sqrt[n]]
%t A260947 Dv[n_]:=Divisors[Prime[n]-1]
%t A260947 Do[Do[Do[If[Mod[k^(Part[Dv[n],t]),Prime[n]]==1,Goto[bb]],{t,1,Length[Dv[n]]-1}];Do[Do[If[Mod[j^(Part[Dv[n],t]),Prime[n]]==1||Mod[(k-j)^(Part[Dv[n],t]),Prime[n]]==1||Mod[((k-j)j*k)^(Part[Dv[n],t]),Prime[n]]==1,Goto[cc]];Continue,{t,1,Length[Dv[n]]-1}];Print[n," ",k];Goto[aa];Label[cc];Continue,{j,1,(k-1)/2}];Label[bb];Continue,{k,1,Prime[n]-1}];Print[n," ",0];Label[aa];Continue,{n,1,60}]
%Y A260947 Cf. A000040, A260946.
%K A260947 nonn
%O A260947 1,5
%A A260947 _Zhi-Wei Sun_, Aug 05 2015