Alphabeta MathHow statement and proof provenance work
The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.
- Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
- AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
- AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.
These labels describe origin, not correctness: citations and verification chips remain separate evidence.
Primitive roots modulo
Definition
Let . A unit is a primitive root modulo when
where the unit group and Euler totient are those of The unit group and Euler's totient for and the order is that of The order of a finite group and the order of an element, with when no positive power of is the identity. Thus the unique class modulo is a primitive root under this definition, since both its order and are .
Depends on
Used by
- 2 is a primitive root modulo 13 by testing the prime divisors of 12 Example
- 2 is primitive modulo every power of 5 Example
- An index table modulo 17 turns multiplication into addition modulo 16 Example
- The primitive roots modulo 17 are 3,5,6,7,10,11,12,14 Example
- For an odd prime p and a primitive root g modulo p, at least one of g and g+p is primitive modulo p² Lemma
- A unit is a primitive root modulo n if and only if it generates (ℤ/nℤ)^× Proposition
- Euler's criterion: if n has a primitive root, gcd(a,n)=1, and m≥1, then xᵐ≡ a (mod n) is solvable if and only if a^φ(n)/gcd(φ(n),m)≡1 (mod n) Theorem
Dependency tree · two levels
18 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
- Peter Hackman, Elementary Number Theory, Chapter C (standard reference, not scraped)
- William Stein, Elementary Number Theory, §2.5 (standard reference, not scraped)