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:

• OEIS 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.

def OEIS.A306477.IsSumOfBinomials (n : ℕ) : Prop

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 $w, x, y, z$.

Equations

• OEIS.A306477.IsSumOfBinomials n = ∃ (w : ℕ) (x : ℕ) (y : ℕ) (z : ℕ), n = (w + 2).choose 2 + (x + 3).choose 4 + (y + 5).choose 6 + (z + 7).choose 8

theorem OEIS.A306477.isSumOfBinomials_1 : IsSumOfBinomials 1

theorem OEIS.A306477.isSumOfBinomials_3 : IsSumOfBinomials 3

theorem OEIS.A306477.isSumOfBinomials_6 : IsSumOfBinomials 6

theorem OEIS.A306477.conjecture (n : ℕ) (hn : 0 < n) : IsSumOfBinomials n

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 $w, x, y, z$.