The 2-4-6-8 Conjecture

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import FormalConjecturesUtil

/-!
# 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).

*References:*
- [A306477](https://oeis.org/A306477)
- [mathoverflow/323541](https://mathoverflow.net/questions/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.
-/

namespace OeisA306477

/-- 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 $w, x, y, z$. -/
def A (n : ℕ) : Prop :=
  ∃ w x y z : ℕ, n = (w + 2).choose 2 + (x + 3).choose 4 + (y + 5).choose 6 + (z + 7).choose 8

@[category test, AMS 11]
theorem a_1 : A 1 :=
  ⟨0, 0, 0, 0, by decide⟩

@[category test, AMS 11]
theorem a_2 : A 2 :=
  ⟨0, 1, 0, 0, by decide⟩

@[category test, AMS 11]
theorem a_3 : A 3 :=
  ⟨1, 0, 0, 0, by decide⟩

@[category test, AMS 11]
theorem a_4 : A 4 :=
  ⟨1, 1, 0, 0, by decide⟩

@[category test, AMS 11]
theorem a_5 : A 5 :=
  ⟨1, 1, 1, 0, by decide⟩

@[category test, AMS 11]
theorem a_6 : A 6 :=
  ⟨2, 0, 0, 0, by decide⟩

/--
**Zhi-Wei Sun's 2-4-6-8 Conjecture (A306477)**: Any integer $n > 0$ can be written as
$\binom{w+2}{2} + \binom{x+3}{4} + \binom{y+5}{6} + \binom{z+7}{8}$ for nonnegative integers $w, x, y, z$.
-/
@[category research open, AMS 11]
theorem conjecture (n : ℕ) (hn : 0 < n) : A n := by
  sorry

end OeisA306477

• Formalization source ↗

On-site verification needs an executable Lean target. None is attached yet.

A source attachment or link is not a verified proof. The statement’s correspondence with the original question needs human review.