# [seqfan] Re: Definition of primitive root (was Re: changes to A205989)

franktaw at netscape.net franktaw at netscape.net
Sun Feb 19 02:45:59 CET 2012

```If you take that definition literally, *every* residue of any integer
that is not a unit modulo that integer is a primitive root. This is
obviously nonsense.

You need to add that 10^(p-1) == 1 (mod p). At which point 10 is not a
primitive root of 2.

Alonso is correct; 10 is not a primitive root of 2.

Franklin T. Adams-Watters

-----Original Message-----
From: Heinz, Alois <alois.heinz at hs-heilbronn.de>

In "pink boxes" of A205989 ...

Alonso del Arte asked: "What is the best way to figure out if a given
prime has primitive root 10?"

David W. Wilson answered: "The best way I know of is to verify that
10^((p-1)/q) != 1 (mod p) for each prime q dividing p-1."

Now take p=2 => p-1=1, and NO prime q divides 1, thus ...
10^((p-1)/q) != 1 (mod p) for each prime q dividing p-1.

So according to this algorithm 10 is a primitive root of 2.

Alois

Am 18.02.2012 23:39, schrieb David Wilson:
> To clarify this to other readers:
>
> A205989(n) gives the smallest prime p >= 10^n with primitive root 10.
> The question is about the value of a(0). Alois contends that 10 is a
> primitive root modulo 2, in which case A205989(0) = 2. I contend that
> 10 is not a primitive root modulo 2, which implies A205989(0) = 7
> (since we would agree that 10 is not a primitive root modulo 3 or 5).
> [ ... ]

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