Deep Learning Architectures for Approximating Goldbach’s Function in New Regions

Abstract

Goldbach conjecture is one of the most famous open mathematical problems. He asserts that: Every even number greater than two is the sum of two prime numbers. The Goldbach function receives an even number and returns the number of different ways to write it as an unordered sum of two prime numbers. We developed a simple multilayer perceptron that attempts to predict Goldbach’s function. This simple model performs well when trained and tested on numbers up to 4 million. However, as expected, the model’s performance significantly deteriorates when trained on smaller numbers (up to 4 million) but tested on larger numbers (4–10 million). To overcome this problem, we present two novel deep learning architectures. In these architectures, we introduce two types of multiplication layers, which we believe are more appropriate for solving mathematical relations. We show that both architectures significantly outperform the simple multilayer perceptron when trained on smaller numbers and tested on larger numbers. We further improve the performance of the deep learning architectures by using a known analytically derived estimation that is used in order to normalize the model’s output.