Decomposing 40 billion integers by four tetrahedral numbers

Based upon a computer search performed on a massively parallel supercomputer, we found that any integer less than billion (B) but greater than 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, 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 compared with that of the old, .