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.

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, 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, 21, 51, 24, 10, 15, 10, 21, 11, 15

Offset: 1

Author

Zhi-Wei Sun, Aug 05 2015

Keywords

• nonn

Comments

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.

Examples

			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.
		

Links

• Zhi-Wei Sun, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000040, A260946.

Programs

Mathematica

SQ[n_]:=IntegerQ[Sqrt[n]]
Dv[n_]:=Divisors[Prime[n]-1]
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}]