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The primitive roots modulo $17$ are $3, 5, 6, 7, 10, 11, 12, 14$

Example

The primitive roots modulo $17$ are

$3, 5, 6, 7, 10, 11, 12, 14.$

Facts & Assumptions

Given: The prime modulus $17$ .

[L1]

A unit is a primitive root modulo $n$ when its order is $φ (n)$ (Primitive roots modulo $n$ ).

[L2]

$φ (17) = 16$ ( $φ (1) = 1$ , and $φ (p) = p - 1$ for every prime $p$ ).

[L3]

If $g$ generates a cyclic group of order $m$ , its generators are $g^{a}$ for the exponent classes coprime to $m$ (The generators of a cyclic group of order $m$ are the $g^{a}$ with $\gcd (a, m) = 1$ , so there are $φ (m)$ of them).

Verification

technique · direct

1.1L1L2algebra

The successive powers of $3$ modulo $17$ for exponents $1$ through $16$ are $3, 9, 10, 13, 5, 15, 11, 16, 14, 8, 7, 4, 12, 2, 6, 1$ , with no earlier $1$ . Thus $3$ has order $16 = φ (17)$ and is primitive by [L1].

2.1step 1.1L3algebra∎

The exponent classes coprime to $16$ are $1, 3, 5, 7, 9, 11, 13, 15$ ; selecting these entries from step 1.1 gives $3, 10, 5, 11, 14, 7, 12, 6$ , which is the displayed set after sorting.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

13 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

William Stein, Elementary Number Theory, Example 2.5.13 (standard reference, not scraped)

The primitive roots modulo 1717 are 3,5,6,7,10,11,12,143,5,6,7,10,11,12,14