A205989 a(n) = smallest prime >= 10^n with primitive root 10.

7, 17, 109, 1019, 10007, 100019, 1000171, 10000019, 100000007, 1000000007, 10000000019, 100000000019, 1000000000061, 10000000000051, 100000000000097, 1000000000000091, 10000000000000061, 100000000000000019, 1000000000000000177, 10000000000000000051

Offset

0,1

Comments

From David W. Wilson, Feb 17 2012 : (Start)

The decimal expansion of 1/a(n) includes every possible block of n digits. Conjecturally, a(n) is the smallest value with this property.

If Artin's conjecture is true, there are an infinite number of primes with primitive root 10, which implies that a(n) exists for all n. Artin's conjecture remains open. (End)

Links

David W. Wilson, Table of n, a(n) for n = 0..75

Dario A. Alpern, Factorization using the Elliptic Curve Method

Maple

with(numtheory):
a:= proc(n) local p;
      p:= nextprime(10^n);
      while 1 in map(q-> 10 &^ ((p-1)/q) mod p, factorset(p-1)) or
            1 <> (10 &^ (p-1) mod p)
      do p:= nextprime(p) od; p
    end:
seq(a(n), n=0..20);  # Alois P. Heinz, Feb 17 2012

Mathematica

spr10[n_]:=Module[{p=NextPrime[n]}, While[PrimitiveRoot[p, 10]!=10, p = NextPrime[ p]]; p]; Join[{7, 17}, Table[spr10[10^d], {d, 2, 20}]] (* Harvey P. Dale, Nov 18 2020 *)

Crossrefs

Sequence in context: A284209 A068172 A067185 * A262474 A284416 A063384

Adjacent sequences: A205986 A205987 A205988 * A205990 A205991 A205992

Keyword

nonn

Author

N. J. A. Sloane, Feb 02 2012

Extensions

More terms from Alois P. Heinz, Feb 17 2012

Status

approved