ExampleConstruction: Literature-sourcedVerification: AI-adaptedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-16

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Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
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The unique cube root of $2$ modulo $29$ is $- 3$

Example

The congruence $x^{3} \equiv 2 (m o d 29)$ has the unique solution class $[- 3]_{29}$ .

Facts & Assumptions

Given: The prime $29$ , exponent $3$ , and target $2$ .

[L1]

If $p$ is prime, $k \geq 1$ , $p ∤ a$ , and $\gcd (k, p - 1) = 1$ , then $x^{k} \equiv a (m o d p)$ has one solution class; if $k ℓ \equiv 1 (m o d p - 1)$ , it is $[a^{ℓ}]_{p}$ (A coprime exponent gives a unique nonzero $k$ -th root modulo a prime).

Verification

technique · direct

1.1

Since $\gcd (3, 28) = 1$ and $3 \cdot 19 = 57 \equiv 1 (m o d 28)$ , [L1] says that the unique cube-root class is $[2^{19}]_{29}$ .

L1givenalgebra

2.1

Modulo $29$ , $2^{8} \equiv 24$ and $2^{10} \equiv 9$ , so $2^{19} = 2^{10} 2^{8} \cdot 2 \equiv 9 \cdot 24 \cdot 2 \equiv 26 \equiv - 3$ ; directly, $(- 3)^{3} = - 27 \equiv 2 (m o d 29)$ .

step 1.1algebra∎

Depends on

A coprime exponent gives a unique nonzero $k$-th root modulo a prime

Used by

Nothing in the library uses this result yet.

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 41 results over 14 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.

Sources

N. Mascot, Legendre Symbols and Quadratic Reciprocity, Chapter 3 (standard reference, not scraped)

The unique cube root of 22 modulo 2929 is −3−3