Erdős sum-product – Less than a week after an AI used algebraic number theory to overturn Paul Erdős’s unit distance problem, Thomas Bloom and colleagues at the University of Manchester have applied a closely related high-dimensional idea to break Erdős’s 1976 sum-product conject
By the time most mathematicians had finished absorbing the shock of AI-driven proof, another crack had already opened in a half-century-old belief.
Just a week after an unreleased OpenAI model disproved an important conjecture first posed by Hungarian mathematician Paul Erdős—called the unit distance problem—Thomas Bloom at the University of Manchester in the UK and his colleagues have now disproved another famous claim that Erdős first posed in 1976. the sum-product conjecture.
Bloom’s path to the new result looks almost like a relay race. The unit-distance breakthrough relied on a trick from algebraic number theory. using an obscure move that creates structures with extremely high dimensions. That high-dimensional construction made it possible to arrange points on a flat surface in a way humans had not considered. and it pushed the number Erdős estimated for the unit distance problem beyond the ceiling many experts had assumed was correct.
Bloom says the key shift came after watching the technique land. “It was a surprise because I had thought about the problem quite a bit,” he said. After seeing what the AI did—using number theory to solve a geometric problem—he and his team realized they could attempt the same strategy for the sum-product conjecture. “Once you know that something might be possible. you’re willing to try a bit harder to actually get it to work. ” Bloom said.
The sum-product conjecture is built around how a set of numbers behaves when you stir its elements in two different ways: addition and multiplication. Erdős’s claim was that if you take a collection of numbers and form new two-number results—one set from sums and another set from products—then at least one of those resulting sets must become much larger than the original. In other words, you can’t keep both outcomes similarly small.
Bloom’s work attacks that promise directly.
Erdős set a bar for how small the larger of the two sets could be. and conjectured it should hold for any set of numbers. Bloom and his colleagues used the same kind of high-dimensional trick to find a set where both the sum set and the multiplied set come in smaller than Erdős thought possible. Instead of relying on the familiar kind of progression—like powers of two. where sums and products behave in starkly different ways—Bloom’s construction works in a more complicated way: it creates a progression of numbers in many different dimensions at the same time. That multi-dimensional setup produces a set where the number of different sums you can make is much smaller.
Bloom described why the breakthrough landed the way it did. “The real surprise for me was that it was so simple,” he said. He added that because the construction can be explained clearly. they now “genuinely understand now why [Erdős’s conjecture] fails. ” which he believes “should help us with lots of other related problems as well.”.
The emotional texture of mathematics here isn’t just about overturning a claim—it’s about how quickly an idea can move once someone sees a door open. Misha Rudnev at the University of Bristol. UK. put it bluntly: “This is typical for maths as a competitive sport.” He said that when a new idea arrives. some people are willing to work “twenty-four hours to find more applications to it. ” and that these are usually the “very good and quick.”.
But Rudnev also pointed to a crucial detail about what’s being broken—and what might still stand.
Erdős’s intuition, Rudnev said, was that the conjecture should mainly be true for integers, or whole numbers. That still appears to be the case. Bloom agreed: “there’s still a huge amount of work to be done; we don’t really understand what’s going on.” The reason the conjecture can fail in Bloom’s construction is that the set used by Bloom and his team relies on exotic number systems that grow more complicated as the sets get larger.
Even with the conjecture fallen, the proof carries a broader message about how problems talk to each other across disciplines. Bloom said the main insight is that questions that look geometric—like sets built from square powers of two—can be tackled with tools from number theory. “It really opens these problems to a whole new community as well,” he said. “People in algebraic number theory weren’t really engaging with these questions.”.
For mathematicians. that’s the tension at the center of the story: in less than a week. an AI technique and a human follow-up have both knocked down long-held barriers. And while the sum-product conjecture is now disproved in its broad form. the hunt is clearly still alive for the version that survives when the numbers are the familiar ones—whole integers—and for the deeper explanation of what changes when the arithmetic landscape gets stranger.
Erdős sum-product conjecture unit distance problem algebraic number theory high-dimensional trick mathematical AI University of Manchester Thomas Bloom Misha Rudnev