Alphabeta Math

Primitive Roots and Unit Groups Modulo N — Examples

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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.

✓ 11 results · all verified · 4 also independently AI-judged
Every result on this page is machine-checked by a proof checker and read in full and owner-audited; the judge is an additional, independent cross-model AI review of the proofs. The 7 not AI-judged were verified by owner audit (typically over a confirmed judge false positive), not failures.
1 · Prerequisites
Abstract Algebra
Combinatorics
Number Theory
Real Analysis
Set Theory
2 · Logical flowchart

3 · Items

Example

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

Example

An index table modulo 17 turns multiplication into addition modulo 16

Example

2 is a primitive root modulo 13 by testing the prime divisors of 12

Example

2 is primitive modulo every power of 5

Example

The unit group modulo 240 decomposes as C₂²× C₄²

Example

λ(561)=80 and every integer coprime to 561 has eightieth power congruent to one

Example

λ(1729)=36 although φ(1729)=1296

Counterexample

(ℤ/15ℤ)^× has order 8 but is not cyclic

Counterexample

φ(8)=4 but every unit modulo 8 has square one

Example

The four square roots of one modulo 128 are 1,63,65,127

Example

The positive moduli below 20 admitting primitive roots are 1,2,3,4,5,6,7,9,10,11,13,14,17,18,19

Sources