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A243755
Primes p such that p is a primitive root modulo the next prime p' and also p' is a primitive root modulo p.
5
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
1,1
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.
LINKS
Zhi-Wei Sun, New observations on primitive roots modulo primes, arXiv:1405.0290 [math.NT], 2014.
EXAMPLE
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.
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}]
CROSSREFS
KEYWORD
nonn
AUTHOR
Zhi-Wei Sun, Jun 09 2014
STATUS
approved