ABSTRACT: The aim of this paper is to study the 3x + 1 problem based on the Collatz iterative formula. It can be seen from the iterative formula that the necessary condition for the Collatz iteration ...

Think about placing dots on a flat surface. You want as many pairs as possible to be separated by the same distance. For any amount of dots, what is the greatest possible number of pairs that can be ...

Artificial intelligence is mastering the kinds of projects that have long helped to build the careers of young mathematicians. What does that mean for their future? Credit...Illustrations by Yoshi ...

For generations, mathematicians have tried and failed to solve an 80-year-old problem. In May, an artificial intelligence (AI) model successfully tackled it. OpenAI, the maker of ChatGPT, said one of ...

Place any number of dots on a two-dimensional plane—say, a piece of paper—and measure the distance between each pair. If you rearrange the dots, how many pairs could be positioned exactly the same ...

In mid-May, OpenAI announced that an internal AI model had disproved the Erdős unit distance conjecture, a famous problem in discrete geometry that had stumped human mathematicians for the last 80 ...

OpenAI's AI model solved the unit distance problem posed by Paul Erdos in 1946 The AI found a counterexample disproving Erdos's conjecture on unit-distance pairs The solution shows unit-distance pairs ...