A260946 - OEIS

%I #5 Aug 05 2015 03:43:15

%S 0,0,0,0,10,0,15,5,5,5,0,17,5,15,5,10,10,26,10,17,5,5,5,5,5,13,15,5,

%T 10,15,15,10,13,13,5,17,5,10,5,10,10,10,25,61,10,13,17,25,5,50,15,13,

%U 17,10,15,5,5,10,17,10,26,10,10,17,5,10,5,5,5,13

%N Least positive integer k < prime(n) such that i^2 + j^2 = k^2 for some 0 < i < j with i*j*k a primitive root modulo prime(n), or 0 if no such k exists.

%C Conjecture: a(n) > 0 for all n > 11. In other words, for any prime p > 31 there are a,b,c among 1,...,p-1 with a^2 + b^2 = c^2 such that a*b*c is a primitive root modulo p.

%H Zhi-Wei Sun, <a href="/A260946/b260946.txt">Table of n, a(n) for n = 1..10000</a>

%e  a(5) = 10 since 10^2 = 6^2 + 8^2, and 6*8*10 = 480 is a primitive root modulo prime(5) = 11.

%t SQ[n_]:=IntegerQ[Sqrt[n]]

%t Dv[n_]:=Divisors[Prime[n]-1]

%t Do[Do[If[SQ[k^2-j^2]==False, Goto[aa]];Do[If[Mod[(Sqrt[k^2-j^2]j*k)^(Part[Dv[n],t]),Prime[n]]==1,Goto[aa]];Continue,{t,1,Length[Dv[n]]-1}];Print[n," ",k];Goto[bb];Label[aa];Continue,{k,1,Prime[n]-1},{j,1,k-1}];Print[n," ",0];Label[bb];Continue,{n,1,70}]

%Y Cf. A000040, A260911, A260947.

%K nonn

%O 1,5

%A _Zhi-Wei Sun_, Aug 05 2015