Diophantine imaging reveals the broken symmetry of sums of integer cubes

Abstract

We introduced a novel method for visualizing large diophantine datasets and in particular found that mapping the known integer triplets $$\{a,b,c\}$$ { a , b , c } solving either equations of the type $$a^3+b^3+c^3=d$$ a 3 + b 3 + c 3 = d or $$a^3+b^3+c^3=d^3$$ a 3 + b 3 + c 3 = d 3 on certain proper subgroups of the circle group exposed a very clear breaking in their symmetry and a strongly non-ergodic distribution of the solutions of sums of three cubes that had never been described before. This method could be further applied to a larger diversity of diophantine problems, informing both number-theoretical conjectures and novel methods in computer sciences on the way, along with paving the road for specific uses of machine learning in exploring diophantine datasets with possible applications in cryptography among others.