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.
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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 pmod n is solvable if and only if a^φ(n)/gcd(φ(n),m)≡1 pmod n Theorem
Dependency tree · next 3 levels
Direct dependencies and their dependencies through the next three levels: 53 results over 15 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
- Peter Hackman, Elementary Number Theory, Chapter C (standard reference, not scraped)
- William Stein, Elementary Number Theory, §2.5 (standard reference, not scraped)