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import FormalConjecturesUtilThe 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).
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.
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, ⊢ 1 = (0 + 2).choose 2 + (0 + 3).choose 4 + (0 + 5).choose 6 + (0 + 7).choose 8 All goals completed! 🐙⟩
@[category test, AMS 11]
theorem a_2 : A 2 :=
⟨0, 1, 0, 0, ⊢ 2 = (0 + 2).choose 2 + (1 + 3).choose 4 + (0 + 5).choose 6 + (0 + 7).choose 8 All goals completed! 🐙⟩
@[category test, AMS 11]
theorem a_3 : A 3 :=
⟨1, 0, 0, 0, ⊢ 3 = (1 + 2).choose 2 + (0 + 3).choose 4 + (0 + 5).choose 6 + (0 + 7).choose 8 All goals completed! 🐙⟩
@[category test, AMS 11]
theorem a_4 : A 4 :=
⟨1, 1, 0, 0, ⊢ 4 = (1 + 2).choose 2 + (1 + 3).choose 4 + (0 + 5).choose 6 + (0 + 7).choose 8 All goals completed! 🐙⟩
@[category test, AMS 11]
theorem a_5 : A 5 :=
⟨1, 1, 1, 0, ⊢ 5 = (1 + 2).choose 2 + (1 + 3).choose 4 + (1 + 5).choose 6 + (0 + 7).choose 8 All goals completed! 🐙⟩
@[category test, AMS 11]
theorem a_6 : A 6 :=
⟨2, 0, 0, 0, ⊢ 6 = (2 + 2).choose 2 + (0 + 3).choose 4 + (0 + 5).choose 6 + (0 + 7).choose 8 All goals completed! 🐙⟩
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 := n:ℕhn:0 < n⊢ A n
All goals completed! 🐙
end OeisA306477