Let $\mathbb N=\{0,1,2,\ldots\}$. Recall that the triangular numbers are those natural numbers $$T_x=\frac {x(x+1)}2\quad \text{with}\ x\in\mathbb N.$$ As $T_x=\binom{x+1}2$, Gauss' triangular number theorem (first claimed by Fermat) can be restated as follows: $$\left\{\binom x2+\binom y2+\binom z2:\ x,y,z\in\mathbb N\right\}=\mathbb N.$$

In view of the above, here I pose a conjecture on representations of integers involving binomial coefficients.

**2-4-6-8 Conjecture.** Any positive integer $n$ can be written as
$$\binom{w}2+\binom{x}4+\binom{y}6+\binom{z}8\quad \text{with}\ w,x,y,z\in\{2,3,\ldots\}.$$

Observe that $$\frac12+\frac14+\frac16+\frac18=\frac{25}{24}\approx 1.0416667.$$ I have verified the above conjecture for all $n=1,\ldots,3\times10^7$. For example, $$4655=\binom{85}2+\binom{14}4+\binom 96+\binom 78=\binom{94}2+\binom 74+\binom 96+\binom{11}8$$ and $$192080=\binom{7}2+\binom{26}4+\binom{25}6+\binom{9}8=\binom{414}2+\binom{39}4+\binom 86+\binom{17}8.$$ See http://oeis.org/A306477 for related data. Note that $1061619$ is the first positive integer not representable as $\binom w2+\binom x4+\binom y6+\binom z9$ with $w,x,y,z\in\mathbb N$.

Question:Are there references in the literature to this or similar conjectures? What are some possible lines of attack for such problems?

I also have some other similar conjectures. See http://oeis.org/A306460, http://oeis.org/A306462 and http://oeis.org/A306471. For example, I conjecture that $$\left\{\binom{2x}2+\binom y2+\binom z3:\ x,y,z=1,2,3,\ldots\right\}=\{1,2,3,\ldots\}$$ and $$\left\{2\binom w3+\binom x3+\binom y3+\binom z3:\ w,x,y,z\in\mathbb N\right\}=\mathbb N.$$ The last equality implies Pollock's conjecture which states that any positive integer is the sum of at most five tetrahedral numbers.

Your comments are welcome!

**Edit:** The 2-4-6-8 conjecture has been verified for $n$ up to $5\times10^{11}$ by Yaakov Baruch, and for $n$ up to $2\times10^{11}$ by Max Alekseyev. I'd like to offer 2468 US dollars as the prize for the first correct proof of the 2-4-6-8 conjecture, or 2468 RMB as the prize for the first explicit counterexample.

**Edit (March 12, 2019):** Today Yaakov Baruch reported that he had verified the 2-4-6-8 conjecture for $n$ up to $2\times 10^{12}$ with no counterexample found.

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