Mathematicians Solve 87-Year-Old Problem with AI

Topic: Mathematics

A mathematician used an artificial intelligence model to find a counterexample to the Jacobian conjecture. This problem had gone unsolved for 87 years and was considered one of the most important in mathematics.

A team of mathematicians has made a groundbreaking discovery using artificial intelligence (AI). Levent Alpöge, working at the AI company Anthropic, used their large language model Claude Fable 5 to find a counterexample to the Jacobian conjecture. This problem had been unsolved for 87 years and was considered one of the most important in mathematics.

The Jacobian conjecture is a complex mathematical concept that deals with functions and how they move points in space. It states that if a function has a constant Jacobian determinant, then there should always exist another function that reverses it. This would return all points to their starting positions.

The conjecture was first stated by Czech mathematician Ludwig Kraus in 1884 and later generalized to any number of dimensions by German mathematician Ott-Heinrich Keller in 1939. It was considered so compelling that Fields Medallist Stephen Smale included it in his 1998 list of Mathematical Problems for the Next Century.

Despite many attempts, nobody had been able to prove or disprove the conjecture until now. Alpöge's function is short enough to fit into a single post on X and has provided a counterexample to the Jacobian conjecture.

Why It Matters

This discovery shows how AI can be used to solve complex mathematical problems that have gone unsolved for decades. It also highlights the importance of collaboration between mathematicians and computer scientists in solving these problems.

Key Facts

• The Jacobian conjecture was first stated by Czech mathematician Ludwig Kraus in 1884.
• The conjecture was generalized to any number of dimensions by German mathematician Ott-Heinrich Keller in 1939.
• The conjecture was considered one of the most important mathematical problems for the next century by Fields Medallist Stephen Smale in 1998.
• Levent Alpöge used the AI model Claude Fable 5 to find a counterexample to the Jacobian conjecture.
• The counterexample was found after 87 years of attempts to prove or disprove the conjecture.

Key Terms

Jacobian determinant

A number that measures how much a function moves points in space

Polynomial mappings

Functions made up of polynomials, which are equations with variables and powers

Implications

This discovery shows how AI can be used to solve complex mathematical problems that have gone unsolved for decades. It also highlights the importance of collaboration between mathematicians and computer scientists in solving these problems.