Lemma totient_semiprime_and_sum from the Aether Catalog (Bridges/AlexanderKnotNumberBridge)

Proved

Bridges.AlexanderTorus.totient_semiprime_and_sum

by raver1975 · Sep 18, 2026 · Mathlib c5ea003 (Lean v4.30.0)

aether-catalogbridges

Helper lemma from Bridges.AlexanderTorus.

theorem Bridges.AlexanderTorus.totient_semiprime_and_sum {p q : ℕ} (hp : p.Prime) (hq : q.Prime) (hne : p ≠ q) :
    Nat.totient (p * q) = (p - 1) * (q - 1) ∧
    p + q = p * q + 1 - Nat.totient (p * q)
 := by sorry

Preamble

import Mathlib
import Definitions.Def_Bridges_AlexanderKnotNumberBridge

open Bridges.AlexanderTorus Polynomial Finset

Formal statement

theorem Bridges.AlexanderTorus.totient_semiprime_and_sum {p q : ℕ} (hp : p.Prime) (hq : q.Prime) (hne : p ≠ q) :
    Nat.totient (p * q) = (p - 1) * (q - 1) ∧
    p + q = p * q + 1 - Nat.totient (p * q)
 := by sorry

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