He–Shiue–Venkat decomposition conjecture for three-variable linear Diophantine equations

Let $a,b,c,n$ be positive integers with $\gcd(a,b,c)=1$. For $i=1,2,3$, let $\hat{S_i}$ be the set of non-negative integer solutions $(x,y,z)$ of $ax+by+cz=n$ satisfying, respectively, $x=0$, $y=0$, and $z=0$.

He–Shiue–Venkat conjecture. For any solution $(\hat{x},\hat{y},\hat{z})$ of $ax+by+cz=n$, there exist $s_i\in\hat{S_i}$ such that

$$(\hat{x},\hat{y},\hat{z})=s_1-s_2+s_3.$$

The paper reports that He, Shiue and Venkat verified the conjecture for some basic examples, but uses its bounds to disprove it; hence the conjecture is refuted.