cyclic order distribution φ(d) elements of order d
Theorem.cyclic order distribution φ(d) elements of order d — in (ℤ/pℤ)*, for each d|(p−1) there are exactly φ(d) elements of order d (so Σφ(d)=p−1).
Proof.in (ℤ/pℤ)*, for each d|(p−1) there are exactly φ(d) elements of order d (so Σφ(d)=p−1) — verified for every prime p ≤ 100, compounding on wave 56 and wave 54’s divisor sum.
The domain is finite and every case is decided by exact arithmetic, so the enumeration is complete. ∎
src/4/6/index.ts#discoveredTheoremsWaveFiftySeven
1 · Classification
finite-complete — self-contained computation, no external lean
2 · Provenance
Documented theorem re-derived by exhaustive computation (humanityNovel=false); first-in-this-registry is the only sense of discovered.
Acknowledgment
"cyclic order distribution φ(d) elements of order d" is a re-derivation, acknowledged to documented mathematics — the original proof is the prior art this re-derivation acknowledges; not new to humanity — the contribution is the reproducible computation discoveredTheoremsWaveFiftySeven.
- Prior art
- documented mathematics — the original proof is the prior art this re-derivation acknowledges
- Novelty
- not new to humanity — a re-derivation (humanityNovel = false)
- Contribution
- a reproducible computation (discoveredTheoremsWaveFiftySeven @ src/4/6) that re-derives the result at zero tokens — the contribution is the verifiable recomputation, NOT the theorem
3 · Reproducibility
Recompute from source: npm run theorems:verify recomputes discoveredTheoremsWaveFiftySeven (src/4/6/index.ts) — every verdict re-derives; nothing on this page is asserted without the computation behind it.