A243837 Positive integers n such that prime(n+i) is a primitive root modulo prime(n+j) for any distinct i and j among 0, 1, 2.

1, 698, 890, 911, 1003, 1141, 1413, 1717, 1807, 1947, 1948, 2216, 2254, 2329, 2455, 2768, 3169, 3224, 3537, 3624, 3737, 3766, 3896, 3904, 3921, 3959, 4027, 4275, 4359, 4427, 4649, 4708, 4845, 5051, 5378, 5386, 5396, 5896, 5897, 6100, 6223, 6226, 6351, 6377

Offset: 1

Author

Zhi-Wei Sun, Jun 11 2014

Keywords

• nonn

Comments

Conjecture: For any integer m > 0, there are infinitely many positive integers n such that prime(n+i) is a primitive root modulo prime(n+j) for any distinct i and j among 0, 1, ..., m.

Examples

			a(1) = 1 since prime(1) = 2 and prime(2) = 3 are primitive roots modulo prime(3) = 5, and 2 and 5 are primitive roots modulo 3, and 3 and 5 are primitive roots modulo 2.
a(2) = 698 since prime(698) = 5261 and prime(699) = 5273 are primitive roots modulo prime(700) = 5279, and 5261 and 5279 are primitive roots modulo 5273, and 5273 and 5279 are primitive roots modulo 5261.
		

Links

• Zhi-Wei Sun, Table of n, a(n) for n = 1..1000
• Zhi-Wei Sun, New observations on primitive roots modulo primes, arXiv:1405.0290 [math.NT], 2014.

Crossrefs

Cf. A000040, A243755, A243839.

Programs

Mathematica

dv[n_]:=Divisors[n]
m=0;Do[Do[If[Mod[Prime[n+1]^(Part[dv[Prime[n]-1],j]),Prime[n]]==1||Mod[Prime[n+2]^(Part[dv[Prime[n]-1],j]),Prime[n]]==1,Goto[aa]],{j,1,Length[dv[Prime[n]-1]]-1}];Do[If[Mod[Prime[n]^(Part[dv[Prime[n+1]-1],i]),Prime[n+1]]==1||Mod[Prime[n+2]^(Part[dv[Prime[n+1]-1],i]),Prime[n+1]]==1,Goto[aa]],{i,1,Length[dv[Prime[n+1]-1]]-1}];Do[If[Mod[Prime[n]^(Part[dv[Prime[n+2]-1],j]),Prime[n+2]]==1||Mod[Prime[n+1]^(Part[dv[Prime[n+2]-1],j]),Prime[n+2]]==1,Goto[aa]],{j,1,Length[dv[Prime[n+2]-1]]-1}];m=m+1;Print[m," ",n];Label[aa];Continue,{n,1,7990}]