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.
Best Answer
2-4-6-8, this we don't appreciate!
There is a long tradition of exploring additive representations of integers by polynomial sequences. The gold standard for measuring such progress is Waring's problem, but the circle method of Hardy and Littlewood to understand such problems works in many related situations. In particular, one could use the circle method to flesh out a conjectural asymptotic formula for problems such as the one in this question. Many of these problems, including the current one, are beyond the reach of current analytic machinery.
To gain a sense of what is feasible, let us return to Waring's problem. If one wishes to express $n$ as a sum of $s$ $k$-th powers, then if there are no local obstructions one expects this to be possible if $s \ge k+1$. However the circle method has any hope of working only if $s \ge 2k+1$ -- this is because even if one had square-root cancelation in the exponential sums over minor arcs, then one would need $s>2k$ for the main term to dominate the error terms. In reality, we are far from this barrier, and one can only prove that $s\ge (1+o(1)) k\log k$ variables suffice. A key exception is the situation of sums of squares -- plain vanilla circle method needs five variables, the Kloostermann refinement of the circle method allows for sums of four variables, and the connection with modular forms/theta functions allows for three variables. The case of squares, where we can do better than the limitations of the circle method, is very special.
Now for a variant of Waring's problem more closely connected to the present question. Roth considered first the problem of writing numbers as the sum of a square, a cube, a fourth power and so on. Roth established that almost all integers up to $x$ may be written as a sum of a square, a cube and a fourth power. This is the best possible result of its kind, because $1/2+ 1/3 < 1 < 1/2+ 1/3 +1/4$. If one asks for all large numbers being represented by an expression of this kind, then the best result known is due to Ford, who showed that all large $n$ are of the form $$ \sum_{i=2}^{k} x_i^i, $$ with $k=15$ being permissible. To compare this with the limitation on the circle method in Waring's problem mentioned above, the relevant quantity is $1/2+1/3+ \ldots +1/k$, and whether this is $>2$. It must be clearly be $>1$ for there to be solutions for all large $n$, and square-root cancelation in minor arc exponential sums would solve the problem if the sum of reciprocals exceeds $2$. This barrier would be crossed at $k=11$, and this gives some rough sense of Ford's work.
The problem here is of the flavor of writing $n$ as $$ \sum_{i=1}^k x_i^{2i}, $$ which is even worse than above. The threshold $\sum_{i=1}^{k} 1/(2i) >2$ is crossed when $k=31$ -- that is, the 2-4-6-8-10-12-...-62 conjecture is the most that circle method aficionados would dream of!
A more reasonable problem would be to ask for almost all integers being represented, in the spirit of Roth. See work of Laporta and Wooley in this (general) context, which will provide further references.