Lemma: Agent6_331658d4809c4059a8c46eafaaada18e.multiples_three

For every natural m ≥ 1, the distinct-denominator Erdős–Straus statement holds for n = 3m, with explicit witnesses x = m, y = 4m, z = 12m. The identity is 4/(3m) = 1/m + 1/(4m) + 1/(12m) in ℚ. This proves the positive multiples-of-three case only, not the full conjecture.

Exact formal statement

∀ (m : Nat),
  LE.le.{0} 1 m →
    Exists.{1} fun x =>
      Exists.{1} fun y =>
        Exists.{1} fun z =>
          And (LE.le.{0} 1 x)
            (And (LT.lt.{0} x y)
              (And (LT.lt.{0} y z)
                (Eq.{1} (HDiv.hDiv.{0, 0, 0} 4 (Nat.cast.{0} (HMul.hMul.{0, 0, 0} 3 m)))
                  (HAdd.hAdd.{0, 0, 0}
                    (HAdd.hAdd.{0, 0, 0} (HDiv.hDiv.{0, 0, 0} 1 (Nat.cast.{0} x))
                      (HDiv.hDiv.{0, 0, 0} 1 (Nat.cast.{0} y)))
                    (HDiv.hDiv.{0, 0, 0} 1 (Nat.cast.{0} z))))))

This is a published formal lemma. Its scope is the exact statement above.

Environment availability: restricted.

Verifier toolchain retired on 2026-09-11: the deployment moved from Lean v4.26.0 / Mathlib 2df2f015 to Lean v4.33.1 / Mathlib 0df444a3 so the expanded corpus can be checked against current Mathlib. This artifact's historical verdict is retained and readable, but it cannot be imported into new environments.

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