306477.lean`

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

8 declarations, 1 contain sorry, 0 declared as axioms

The 2-4-6-8 Conjecture

Any integer $n > 0$ can be written as $\binom{w+2}{2} + \binom{x+3}{4} + \binom{y+5}{6} + \binom{z+7}{8}$ with $w, x, y, z$ nonnegative integers.

Zhi-Wei Sun has offered a $2,468 prize for the first proof (or $2,468 RMB for a counterexample).

The conjecture has been verified for all $n$ up to $1.2 \times 10^{12}$ by Yaakov Baruch (March 2019).

The conjecture is false: $n = 896315812331399$ has no such representation. The counterexample was found in the OEIS Open project, with a Lean proof.

References:

A306477
mathoverflow/323541: Z.-W. Sun, "Positive integers written as C(w,2) + C(x,4) + C(y,6) + C(z,8) with w,x,y,z in {2,3,...},", Feb. 19, 2019.
T. Adamczewski, "OEIS Open: How many conjectures can language models turn into theorems?", arXiv:2608.11941 [cs.AI], 2026.

Imports (1)

FormalConjecturesUtil

def OeisA306477.A The predicate that n can be written as $\binom{w+2}{2} + \binom{x+3}{4} + \binom{y+5}{6} + \binom{z+7}{8}$ for nonnegative integers $w, x, y, z$ line 46 source
Source documentation
The predicate that n can be written as $\binom{w+2}{2} + \binom{x+3}{4} + \binom{y+5}{6} + \binom{z+7}{8}$ for nonnegative integers $w, x, y, z$.

no axioms what this means
Statement uses: Nat
Proof uses: Exists, Eq, HAdd.hAdd, Nat.choose, OfNat.ofNat
3 instances, coercions and auxiliary constants left out.
```
def A (n : ℕ) : Prop :=
…
```
theorem OeisA306477.a_1 line 50 source
no axioms what this means
Statement uses: OeisA306477.A, OfNat.ofNat, Nat
Proof uses: Exists.intro, Exists, Eq, HAdd.hAdd, Nat.choose, of_decide_eq_true, id, Bool, Decidable.decide, Bool.true, Eq.refl
5 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
theorem OeisA306477.a_2 line 54 source
no axioms what this means
Statement uses: OeisA306477.A, OfNat.ofNat, Nat
Proof uses: Exists.intro, Exists, Eq, HAdd.hAdd, Nat.choose, of_decide_eq_true, id, Bool, Decidable.decide, Bool.true, Eq.refl
5 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
theorem OeisA306477.a_3 line 58 source
no axioms what this means
Statement uses: OeisA306477.A, OfNat.ofNat, Nat
Proof uses: Exists.intro, Exists, Eq, HAdd.hAdd, Nat.choose, of_decide_eq_true, id, Bool, Decidable.decide, Bool.true, Eq.refl
5 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
theorem OeisA306477.a_4 line 62 source
no axioms what this means
Statement uses: OeisA306477.A, OfNat.ofNat, Nat
Proof uses: Exists.intro, Exists, Eq, HAdd.hAdd, Nat.choose, of_decide_eq_true, id, Bool, Decidable.decide, Bool.true, Eq.refl
5 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
theorem OeisA306477.a_5 line 66 source
no axioms what this means
Statement uses: OeisA306477.A, OfNat.ofNat, Nat
Proof uses: Exists.intro, Exists, Eq, HAdd.hAdd, Nat.choose, of_decide_eq_true, id, Bool, Decidable.decide, Bool.true, Eq.refl
5 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
theorem OeisA306477.a_6 line 70 source
no axioms what this means
Statement uses: OeisA306477.A, OfNat.ofNat, Nat
Proof uses: Exists.intro, Exists, Eq, HAdd.hAdd, Nat.choose, of_decide_eq_true, id, Bool, Decidable.decide, Bool.true, Eq.refl
5 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
theorem OeisA306477.conjecture sorry line 83 source
depends on sorry what this means
Statement uses: Not, Nat, LT.lt, OfNat.ofNat, OeisA306477.A
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
4 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.