Decomposing 40 billion integers by four tetrahedral numbers

Based upon a computer search performed on a massively parallel supercomputer, we found that any integer $n$ less than $40$ billion ($40$B) but greater than $343,867$ can be written as a sum of four or fewer tetrahedral numbers. This result has established a new upper bound for a conjecture compared to an older one, $1$B, obtained a year earlier. It also gives more accurate asymptotic forms for partitioning. All this improvement is a direct result of algorithmic advances in efficient memory and cpu utilizations. The heuristic complexity of the new algorithm is $O(n)$ compared with that of the old, $O(n^{5/3}\log n)$.