Just a week. That’s how long it took for a 50-year-old math problem to collapse after an AI took down its older, 80-year cousin.
Last week, OpenAI’s unreleased model shocked everyone. It disproved the unit distance problem. A conjecture proposed by Hungarian mathematician Paul Erdős. He called it his “most striking contribution to geometry.” Experts spent decades failing to crack it. The question? How many similar-sized connections can you draw between dots on a flat plane?
Erdős thought he found the ceiling. The community agreed.
The AI didn’t.
Using obscure algebraic number theory, it built structures in high dimensions. Structures humans never considered. The arrangement of dots shifted into something unrecognizable. The number of connections exploded upward. Mathematicians were stunned. Some thought Erdős’s conjecture was safe until the day they died.
Once you know that something might be possible you’re willing to try a bit.
Thomas Bloom at the University of Manchester heard about the AI’s hack. He saw the pattern. Number theory solving geometry? Counter-intuitive. But potent. He and his colleagues turned that same logic onto Erdős’s other famous claim. The sum-product conjecture. Posted in 1976. Still standing until Tuesday.
What is the sum-product conjecture?
Imagine a set of numbers. You add every pair together. Then you multiply every pair together. Two new sets emerge. Erdős bet that one of these new sets has to be massive. Much bigger than the original. You can’t have both stay small.
Think 1 through 5. Multiplication wins. 1+4 and 2+3 both equal 5. Duplicates shrink the sum set. Multiplication scatters everything wider.
Switch the input. Use powers of two. 1, 2, 04, 08, 16. Now addition creates distinct totals. Multiplication just makes more powers. The sum set balloons.
Erdős set the bar for how “small” the larger set could possibly be. He claimed this held for any set.
Bloom broke the bar.
Not with complex geometry. With high dimensions. By creating a numerical progression across multiple dimensions simultaneously, he found a set where both the sums and the products stayed small. Smaller than Erdős allowed.
“The construction is so simple,” Bloom admitted.
Really simple.
Misha Rudnev from the University of Bristol called it a contact sport. When a new idea drops, good mathematicians run on zero sleep. They find applications fast. Rudnev notes that Erdős originally guessed this logic worked for integers. Whole numbers.
Does the human proof hold up there?
Yes. Bloom agrees. The exotic number system his team used gets wildly complicated as the set grows. Integers are still safe. The mystery remains intact for standard math. But the general conjecture? Dead.
The real takeaway isn’t the answer. It’s the door the AI opened. Problems looking geometric might actually need number theory.
Who solves geometry? Geometers. Who solves number theory? Algebraists. They rarely talk. Now they have to.
A whole new community just woke up. And the math hasn’t finished breaking yet.
