Peculiarities of the Operation of Complex Computational Algorithms using the Example of Testing the Polock Hypothesis

The aim of this article is to investigate and experimentally verify Pollock's third hypothesis, which states that any natural number can be represented as the sum of no more than five tetrahedral numbers. The main focus is on developing an efficient algorithm capable of processing large numbers to test the hypothesis on a numerical range up to one billion. Within the research, an algorithm was designed and implemented that combines several modern optimization techniques: two-pointer search, approximation methods, multithreading, and memorization testing. Particular attention was paid to benchmarking different implementations across platforms — NodeJS, Bun, C, Python and GPU. The test results are presented in the form of graphs for both single-threaded and multi-threaded code execution. Experimental results showed that Bun in multithreaded mode demonstrated the highest performance, even outperforming C in certain cases. This supports the developers’ claims regarding the speed of Bun. A comparison between CPU and GPU implementations using Python revealed that GPU acceleration provides only a slight improvement in performance while shader language significantly increasing code complexity compared to Python code. The experimental study covered all natural numbers up to one billion to verify the hypothesis. As a result of testing, only 241 numbers required five tetrahedral summands, with the last such exceptional number being 343,867. This finding aligns with previous research. The algorithm showed stable memory usage across both small and large numbers. Future studies may focus on utilizing CUDA or high-performance computing clusters to validate the hypothesis over even larger numerical ranges.