1,7

Conjecture: a(n) > 0 for all n > 2. In other words, for any prime p > 3, there exists a positive primitive root g < p modulo p such that 2^g - 1 and (g-1)! are also primitive roots modulo p.

We have verified this for all primes p with 3 < p < 10^6.

Zhi-Wei Sun, Table of n, a(n) for n = 1..1500

Zhi-Wei Sun, Notes on primitive roots modulo primes, arXiv:1405.0290 [math.NT], 2014.

a(4) = 1 since 5, 2^5 - 1 = 31 and (5-1)! = 24 are all primitive roots modulo prime(4) = 7.

a(6) = 1 since 11, 2^(11) - 1 = 2047 and (11-1)! = 3628800 are all primitive roots modulo prime(6) = 13. Note that both 2047 and 3628800 are congruent to 6 modulo 13.

a(14) = 1 since 34, 2^(34) - 1 and (34-1)! are all primititive roots modulo prime(14) = 43. Note that 2^(34) - 1 == 20 (mod 43) and 33! == -14 (mod 43).

f[n_]:=2^n-1

g[n_]:=(n-1)!

rMod[m_, n_]:=Mod[m, n, -n/2]

dv[n_]:=Divisors[n]

Do[m=0; Do[If[Mod[f[k], Prime[n]]==0, Goto[aa]]; Do[If[Mod[k^(Part[dv[Prime[n]-1], i]), Prime[n]]==1||Mod[rMod[f[k], Prime[n]]^(Part[dv[Prime[n]-1], i]), Prime[n]]==1||Mod[rMod[g[k], Prime[n]]^(Part[dv[Prime[n]-1], i]), Prime[n]]==1, Goto[aa]], {i, 1, Length[dv[Prime[n]-1]]-1}]; m=m+1; Label[aa]; Continue, {k, 1, Prime[n]-1}]; Print[n, " ", m]; Continue, {n, 1, 70}]

Cf. A000040, A000142, A000225, A236966, A237112, A242242, A242250.

Sequence in context: A010275 A176494 A157229 * A107297 A107296 A080847

Adjacent sequences: A242245 A242246 A242247 * A242249 A242250 A242251

Zhi-Wei Sun, May 09 2014

approved