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%I #8 Aug 06 2015 04:58:18
%S 0,0,0,0,5,5,5,17,13,17,10,10,5,13,13,25,5,5,39,25,17,5,5,5,17,29,5,5,
%T 5,5,5,5,5,34,17,5,5,26,13,13,5,10,29,13,13,5,34,5,5,5,5,25,25,5,5,13,
%U 17,5,5,10,29,13,13,61,17,13,17,17,5,13
%N Least positive integer k < prime(n) such that there are 0 < i < j with i^2 + j^2 = k^2 for which (i*j)/2 is a primitive root modulo prime(n), or 0 if no such k exists.
%C Conjecture: a(n) > 0 for any n > 4. In other words, for any prime p > 7, there exists a right triangle whose three sides are among 1,...,p-1 and whose area is a primitive root modulo p.
%C We have verified this for primes p < 10^5.
%C We also conjecture that for any prime p > 31, there exists a right triangle whose three sides are among 1,...,p-1, and whose perimeter and area are quadratic residues modulo p.
%H Zhi-Wei Sun, <a href="/A260960/b260960.txt">Table of n, a(n) for n = 1..10000</a>
%e a(7) = 5 since 3^2 + 4^2 = 5^2, and (3*4)/2 = 6 is a primitive root modulo prime(7) = 17.
%t SQ[n_]:=IntegerQ[Sqrt[n]]
%t Dv[n_]:=Divisors[Prime[n]-1]
%t Do[Do[Do[If[SQ[k^2-j^2]==False, Goto[cc]];Do[If[Mod[(j*Sqrt[k^2-j^2]/2)^(Part[Dv[n],t]),Prime[n]]==1,Goto[cc]];Continue,{t,1,Length[Dv[n]]-1}];
%t Print[n," ",k];Goto[aa];Label[cc];Continue,{j,1,k-1}];Label[dd];Continue,{k,1,Prime[n]-1}];Print[n," ",0];Label[aa];Continue,{n,1,70}]
%Y Cf. A000040, A003273, A260911, A260946, A260947.
%K nonn
%O 1,5
%A _Zhi-Wei Sun_, Aug 06 2015
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