|
%I #10 Aug 05 2019 03:03:09
%S 2,3,5,11,59,61,83,101,131,151,179,181,197,251,257,269,271,317,337,
%T 347,367,419,443,461,523,563,577,587,593,659,709,733,797,811,821,827,
%U 863,947,971,977,1061,1063,1069,1097,1129,1153,1171,1187,1217,1229,1277,1283,1301,1361,1433,1451,1543,1553,1601,1619
%N Primes p such that p is a primitive root modulo the next prime p' and also p' is a primitive root modulo p.
%C Conjecture: The sequence contains infinitely many primes. Moreover, there are infinitely many primes p such that both p and -p are primitive roots modulo the next prime p' and both p' and -p' are primitive roots modulo p.
%H Zhi-Wei Sun, <a href="/A243755/b243755.txt">Table of n, a(n) for n = 1..10000</a>
%H 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.
%e a(1) = 2 since prime(1) = 2 is a primitive root modulo prime(2) = 3 and also prime(2) = 3 is a primitive root modulo prime(1) = 2.
%e a(2) = 3 since prime(2) = 3 is a primitive root modulo prime(3) = 5 and also prime(3) = 5 is a primitive root modulo prime(2) = 3.
%t dv[n_]:=Divisors[n]
%t n=0;Do[Do[If[Mod[(Prime[m])^(Part[dv[Prime[m+1]-1],i]),Prime[m+1]]==1,Goto[aa]],{i,1,Length[dv[Prime[m+1]-1]]-1}];Do[If[Mod[Prime[m+1]^(Part[dv[Prime[m]-1],j]),Prime[m]]==1,Goto[aa]],{j,1,Length[dv[Prime[m]-1]]-1}];n=n+1;Print[n," ",Prime[m]];Label[aa];Continue,{m,1,256}]
%Y Cf. A000040, A243164, A243403.
%K nonn
%O 1,1
%A _Zhi-Wei Sun_, Jun 09 2014
|
