2326.lean`

Explanation legend Lean4 Domain Explanation Language (L4DEL)

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L4DEL: Lean4 Domain Explanation Language

Annotation provenance. This overlay contains 10 curated AI-written annotations and 7190 mechanical source-derived annotations. They have not been human-reviewed as explanations; treat them as unverified and not mathematical authority. Retained generation history: The 71 original curated annotations were produced in a Claude Sonnet 5.5 session at medium effort; Claude Opus 5.5 is recorded in that session as the advisor model, not the direct author. The bulk source-derived expansion was generated mechanically from pinned source comments by tooling developed in a GPT-5.6 Sol session at ultra reasoning; GPT-5.6 Sol did not originate the quoted source prose. On 2026-10-17 the mechanically generated source-comment annotations of the Formal Conjectures, Carleson, FLT, PFR, CompFiles, Mathematics in Lean and Theorem Proving in Lean overlays were regenerated by the same generator after three corrections made in a Claude Sonnet 5.5 session: titles no longer cut a sentence inside mathematics or end on a dangling word, workflow markers are removed from the quoted prose, and declaration names follow the scanner's namespace resolution. The boilerplate step line was dropped, and comments that were only a row of dashes no longer yield annotations. No new prose was written and no human review is claimed.

Credit: The upstream authors and contributors retain credit for the project and mathematics. lean4-explainer only renders their pinned source and adds the separately disclosed explanation layer.

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

8 declarations, 2 contain sorry, 0 declared as axioms

Multiplicative order of 2 mod $2n+1$

The multiplicative order of 2 modulo $2n+1$. In other words, the least $m > 0$ such that $2n+1$ divides $2^m - 1$.

References:

A002326

Imports (1)

FormalConjecturesUtil

def OeisA2326.a The multiplicative order of 2 modulo $2n+1$ line 34 source
Source documentation
The multiplicative order of 2 modulo $2n+1$.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat
Proof uses: orderOf, ZMod, HAdd.hAdd, HMul.hMul, OfNat.ofNat
15 instances, coercions and auxiliary constants left out.
```
noncomputable def a (n : ℕ) : ℕ :=
…
```
theorem OeisA2326.a_0 Value of the sequence a at 0 line 39 source
Source documentation
Value of the sequence a at 0.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA2326.a, OfNat.ofNat
Proof uses: id, Subsingleton.orderOf_eq, ZMod
15 instances, coercions and auxiliary constants left out.

Tactic trace: 2 steps
Open to load the checked before-and-after goal states.
```
theorem a_0 : a 0 = 1 := by
…
```
Mentions: OeisA2326.a
theorem OeisA2326.a_1 Value of the sequence a at 1 line 45 source
Source documentation
Value of the sequence a at 1.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA2326.a, OfNat.ofNat
Proof uses: id, Eq.mpr, orderOf, ZMod, HAdd.hAdd, HMul.hMul, And, HPow.hPow, LT.lt, Ne, congrArg, propext, orderOf_eq_iff, of_decide_eq_true, Bool, Decidable.decide, Bool.true, Eq.refl, And.intro, Eq.symm and 8 more
31 instances, coercions and auxiliary constants left out.

Tactic trace: 7 steps
Open to load the checked before-and-after goal states.
```
theorem a_1 : a 1 = 2 := by
…
```
Mentions: OeisA2326.a
theorem OeisA2326.a_2 Value of the sequence a at 2 line 52 source
Source documentation
Value of the sequence a at 2.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA2326.a, OfNat.ofNat
Proof uses: id, Eq.mpr, orderOf, ZMod, HAdd.hAdd, HMul.hMul, And, HPow.hPow, LT.lt, Ne, congrArg, propext, orderOf_eq_iff, of_decide_eq_true, Bool, Decidable.decide, Bool.true, Eq.refl, And.intro, dite and 11 more
33 instances, coercions and auxiliary constants left out.

Tactic trace: 7 steps
Open to load the checked before-and-after goal states.
```
theorem a_2 : a 2 = 4 := by
…
```
Mentions: OeisA2326.a
theorem OeisA2326.a_3 Value of the sequence a at 3 line 59 source
Source documentation
Value of the sequence a at 3.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA2326.a, OfNat.ofNat
Proof uses: id, Eq.mpr, orderOf, ZMod, HAdd.hAdd, HMul.hMul, And, HPow.hPow, LT.lt, Ne, congrArg, propext, orderOf_eq_iff, of_decide_eq_true, Bool, Decidable.decide, Bool.true, Eq.refl, And.intro, dite and 11 more
33 instances, coercions and auxiliary constants left out.

Tactic trace: 7 steps
Open to load the checked before-and-after goal states.
```
theorem a_3 : a 3 = 3 := by
…
```
Mentions: OeisA2326.a
theorem OeisA2326.a_4 Value of the sequence a at 4 line 66 source
Source documentation
Value of the sequence a at 4.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA2326.a, OfNat.ofNat
Proof uses: id, Eq.mpr, orderOf, ZMod, HAdd.hAdd, HMul.hMul, And, HPow.hPow, LT.lt, Ne, congrArg, propext, orderOf_eq_iff, of_decide_eq_true, Bool, Decidable.decide, Bool.true, Eq.refl, And.intro, dite and 11 more
33 instances, coercions and auxiliary constants left out.

Tactic trace: 7 steps
Open to load the checked before-and-after goal states.
```
theorem a_4 : a 4 = 6 := by
…
```
Mentions: OeisA2326.a
theorem OeisA2326.conjecture1 If $p$ is an odd prime then $a((p^3-1)/2) = p \cdot a((p^2-1)/2)$ sorry line 78 source
Source documentation
If $p$ is an odd prime then $a((p^3-1)/2) = p \cdot a((p^2-1)/2)$. Because otherwise $a((p^3-1)/2) < p \cdot a((p^2-1)/2)$ iff $a((p^3-1)/2) = a((p-1)/2)$ for a prime $p$. Equivalently $p^3$ divides $2^{p-1}-1$, but no such prime $p$ is known.
- Thomas Ordowski, Feb 10 2014
depends on sorry what this means
Statement uses: Nat, Nat.Prime, Ne, OfNat.ofNat, Eq, OeisA2326.a, HDiv.hDiv, HSub.hSub, HPow.hPow, HMul.hMul
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 conjecture1 (p : ℕ) (hp : p.Prime) (hp_odd : p ≠ 2) :
    a ((p ^ 3 - 1) / 2) = p * a ((p ^ 2 - 1) / 2) := by
…
```
Mentions: OeisA2326.a
theorem OeisA2326.conjecture2 A generalization of the previous conjecture: For each $k \ge 2$, if $p$ is an odd prime then $a((p^{k+1}-1)/2) = p \cdot a((p^k-1)/2)$ sorry line 96 source
Source documentation
A generalization of the previous conjecture: For each $k \ge 2$, if $p$ is an odd prime then $a((p^{k+1}-1)/2) = p \cdot a((p^k-1)/2)$. Computer testing of this generalized conjecture shows that there is no counterexample for $k$ and $p$ both up to 1000.
- Ahmad J. Masad, Oct 17 2020
The two are in fact equivalent. Its $k = 2$ instance is conjecture1. Conversely, write $d = \operatorname{ord}_{p^2}(2)$. conjecture1 for $p$ says $v_p(2^d - 1) = 2$, and lifting the exponent for the odd prime $p$ then gives $v_p(2^{d p^t} - 1) = 2 + t$, hence $\operatorname{ord}_{p^{2+t}}(2) = d p^t$ for every $t \ge 0$, which is every instance for that $p$.
L4DEL explanation
A generalization of the previous conjecture: For each $k \ge 2$, if $p$ is an odd prime then $a((p^{k+1}-1)/2) = p \cdot a((p^k-1)/2)$. Computer testing of this generalized conjecture shows that there is no counterexample for $k$ and $p$ both up to 1000. - Ahmad J. Masad, Oct 17 2020 The two are in fact equivalent. Its $k = 2$ instance is conjecture1. Conversely, write $d = \operatorname{ord}_{p^2}(2)$. conjecture1 for $p$ says $v_p(2^d - 1) = 2$, and lifting the exponent for the odd prime $p$ then gives $v_p(2^{d p^t} - 1) = 2 + t$, hence $\operatorname{ord}_{p^{2+t}}(2) = d p^t$ for every $t \ge 0$, which is every instance for that $p$.
Sources and attribution: Pinned Lean source (1646ca16afd6)

depends on sorry what this means
Statement uses: Nat, LE.le, OfNat.ofNat, Nat.Prime, Ne, Eq, OeisA2326.a, HDiv.hDiv, HSub.hSub, HPow.hPow, HAdd.hAdd, HMul.hMul
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
28 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem conjecture2 (k : ℕ) (hk : 2 ≤ k) (p : ℕ) (hp : p.Prime) (hp_odd : p ≠ 2) :
    a ((p ^ (k + 1) - 1) / 2) = p * a ((p ^ k - 1) / 2) := by
…
```
Mentions: OeisA2326.a