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 ...

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BERLIN — The collapse of a €100 billion program to build a European next-generation jet fighter has countries scrambling for options. For nearly a decade, the Future Combat Air System was supposed to ...

The result is correct but challenges core norms of mathematics: checking proofs, crediting ideas and keeping research open to everyone.

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 ...

Mathematician Will Sawin discusses his experience reviewing and refining a mathematical proof devised by OpenAI's internal model—and what that could mean for mathematics. Reading time 10 minutes Will ...

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 ...

Richard Feynman described a decision-making problem and its solution in handwritten notes, but the meaning of the notes has been a mystery for almost 50 y. We decipher the problem and solution from ...

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 ...