TITLE: Waring problem for binomial coefficients (generalization of Gauss' Eureka Theorem)
QUESTION [6 upvotes]: Is there a number k such that every natural number can be written as $\sum_{i=1}^k \binom{a_i}{3}$ for some natural numbers $a_i$'s?

REPLY [2 votes]: In more general case this problem was studied by Nechaev, see 

On the question of representing natural numbers by a~sum of terms of the form $x(x+1)\ldots (x+n-1)/n!$ 
On the representation of natural numbers as a sum of terms of the form $x(x+1)\ldots (x+n-1)/n!$