Shortly after an artificial intelligence system from OpenAI disproved an 80-year-old mathematical conjecture, researchers have applied similar methods to resolve another longstanding problem that had remained unsolved for fifty years.
The recent AI achievement involved disproving a conjecture by Hungarian mathematician Paul Erdős on unit distances. The model used an advanced technique from algebraic number theory to demonstrate that certain arrangements of points on a plane could exceed previously assumed limits.
Building on this approach, Thomas Bloom of the University of Manchester and collaborators have now disproved Erdős’s sum-product conjecture from 1976. The conjecture proposed that for any set of numbers, either the sums or the products of pairs would produce a significantly larger collection than the original set.
Bloom’s team employed high-dimensional constructions to identify sets where both sums and products remained smaller than Erdős had predicted. The method revealed that problems appearing geometric can be addressed through tools from number theory.
Experts note that the original conjecture may still apply to ordinary integers, though the new counterexamples involve more complex number systems. The work highlights how recent ideas can quickly lead to further applications across related mathematical questions.
