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$2$ is primitive modulo every power of $5$

Example

For every $k \geq 1$ , the class of $2$ is a primitive root modulo $5^{k}$ .

Facts & Assumptions

Given: The integer $2$ and powers of the odd prime $5$ .

[L1]

For an odd prime $p$ , an integer $u$ with $p ∤ u$ , and $k \geq 1$ , the class of $1 + p u$ has order $p^{k - 1}$ modulo $p^{k}$ (For odd prime $p$ , $p ∤ u$ , and $k \geq 1$ , the class of $1 + p u$ has order $p^{k - 1}$ modulo $p^{k}$ ).

[L2]

If an element has order $r$ , then its fourth power has order $r / \gcd (r, 4)$ (In a cyclic group of order $m$ , $g^{a}$ has order $m / \gcd (a, m)$ ).

[L3]

An element of order $r$ has its $t$ th power equal to the identity exactly when $r ∣ t$ (If $ord (g) = n$ then $g^{k} = e$ iff $k$ is an integer multiple of $n$ , the powers $g^{0}, \dots, g^{n - 1}$ are distinct, and $⟨ g ⟩$ has exactly $n$ elements; if $g$ has infinite order then $g^{j} = g^{k}$ only for $j = k$ ).

[L4]

$φ (5^{k}) = 4 \cdot 5^{k - 1}$ (For a prime $p$ and $k \geq 1$ , $φ (p^{k}) = p^{k} - p^{k - 1}$ ).

[L5]

A class modulo $n$ is a unit exactly when its representative is coprime to $n$ , and a primitive root is a unit of order $φ (n)$ (For $n \geq 1$ , $[a]_{n}$ is a unit if and only if $\gcd (a, n) = 1$ , Primitive roots modulo $n$ ).

Verification

technique · direct

1.1L3L4L5algebra

Modulo $5$ , $2^{1}, 2^{2}, 2^{3}, 2^{4}$ are $2, 4, 3, 1$ , so $2$ has order $4$ . This settles $k = 1$ .

1.2L1L2

Let $k \geq 2$ and let $r$ be the order of $2$ modulo $5^{k}$ . Since $2^{4} = 1 + 5 \cdot 3$ , [L1] says that $2^{4}$ has order $5^{k - 1}$ . Hence [L2] gives $r / \gcd (r, 4) = 5^{k - 1}$ .

2.1step 1.1step 1.2L3L4L5∎

Reduction modulo $5$ and step 1.1 show that $4 ∣ r$ by [L3]. Therefore $\gcd (r, 4) = 4$ , and step 1.2 yields $r = 4 \cdot 5^{k - 1} = φ (5^{k})$ by [L4]. Since $2$ is a unit by [L5], it is primitive by the definition in [L5].

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

44 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, Theorem 2.5.11 (standard reference, not scraped)

22 is primitive modulo every power of 55