ScholzConjecture.lean`

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Project sources: Pinned project documentation, Formal Conjectures paper, arXiv v1

3 declarations, 1 contain sorry, 0 declared as axioms

Scholz conjecture on addition chains

References:

Wikipedia
MathWorld
Tall22 Amadou Tall. "The Scholz conjecture on addition chain is true for infinitely many integers with $\ell(2n) = \ell(n)$." arXiv:2210.13812 (2022). Also available as ePrint 2023/020.
OEIS A003313

Imports (1)

FormalConjecturesUtil

theorem ScholzConjecture.scholz_conjecture The Scholz conjecture, also known as the Scholz-Brauer conjecture, asserts that for every positive integer $n$, the addition-chain length of $2^n - 1$ is at most $n - 1 + \ell(n)$ sorry line 44 source
Source documentation
The Scholz conjecture, also known as the Scholz-Brauer conjecture, asserts that for every positive integer $n$, the addition-chain length of $2^n - 1$ is at most $n - 1 + \ell(n)$.

depends on sorry what this means
Statement uses: Iff, True, Nat, LT.lt, OfNat.ofNat, LE.le, additionChainLength, HSub.hSub, HPow.hPow, HAdd.hAdd
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
22 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem scholz_conjecture :
    answer(sorry) ↔ ∀ (n : ℕ), 0 < n → ℓ(2 ^ n - 1) ≤ n - 1 + ℓ(n) := by
…
```
lemma ScholzConjecture.three_notMem_additionChainSteps_seven 7 is the first value where the doubling bound is not sharp: it gives ℓ(7) ≥ 3, and no four-entry chain ends at 7 line 54 source
Source documentation
7 is the first value where the doubling bound is not sharp: it gives ℓ(7) ≥ 3, and no four-entry chain ends at 7. Every entry of such a chain lies strictly between 1 and 7, so there are only finitely many to rule out.
```
private lemma three_notMem_additionChainSteps_seven : 3 ∉ additionChainSteps 7 := by
…
```
Mentions: additionChainSteps
theorem ScholzConjecture.additionChainLength_first_values The first few values of $\ell(n)$ line 68 source
Source documentation
The first few values of $\ell(n)$. See OEIS A003313.

L4DEL explanation
The first few values of $\ell(n)$. See OEIS A003313.
Sources and attribution: Pinned Lean source (1646ca16afd6)

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, List, Nat, List.cons, additionChainLength, OfNat.ofNat, List.nil
Proof uses: Iff.mp, LE.le, Nat.le_zero, additionChainLength_le, of_decide_eq_true, IsAdditionChain, id, Bool, Decidable.decide, Bool.true, Eq.refl, rfl, Option, List.getLast?, List.length, Option.some, HAdd.hAdd, LT.lt, HPow.hPow, HSub.hSub and 25 more
29 instances, coercions and auxiliary constants left out.

Tactic trace: 49 steps
Open to load the checked before-and-after goal states.
```
theorem additionChainLength_first_values :
    [ℓ(1), ℓ(2), ℓ(3), ℓ(4), ℓ(5), ℓ(6), ℓ(7), ℓ(8), ℓ(9), ℓ(10)] =
    [0, 1, 2, 2, 3, 3, 4, 3, 4, 4] := by
…
```