A243755 Primes p such that p is a primitive root modulo the next prime p' and also p' is a primitive root modulo p.

2, 3, 5, 11, 59, 61, 83, 101, 131, 151, 179, 181, 197, 251, 257, 269, 271, 317, 337, 347, 367, 419, 443, 461, 523, 563, 577, 587, 593, 659, 709, 733, 797, 811, 821, 827, 863, 947, 971, 977, 1061, 1063, 1069, 1097, 1129, 1153, 1171, 1187, 1217, 1229, 1277, 1283, 1301, 1361, 1433, 1451, 1543, 1553, 1601, 1619

Offset: 1

Author

Zhi-Wei Sun, Jun 09 2014

Keywords

• nonn

Comments

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.

Examples

			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.
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.
		

Links

• Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
• Zhi-Wei Sun, New observations on primitive roots modulo primes, arXiv:1405.0290 [math.NT], 2014.

Crossrefs

Cf. A000040, A243164, A243403.

Programs

Mathematica

dv[n_]:=Divisors[n]
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}]