An AI named Aristotle just finished a mathematical proof that had stumped humans for decades and then verified its own work in a formal coding language.
Number theorists have long struggled to understand the specific gaps between numbers in Sidon sets where every pair has a unique product. A new AI system independently discovered the proof for a long-standing conjecture about these sets. Not only did the machine find the logic, but it also wrote the proof in Lean, a language used to formally verify that mathematical arguments are 100 percent correct. This marks a shift from machines helping humans to machines conducting the entire cycle of discovery and verification. It suggests a future where the hardest problems in math are solved by silicon brains while we simply check the results.
Gaps in Multiplicative Sidon Sets
arXiv · 2605.02064
For a positive integer $n$, let $g(n)$ denote the infimum of all real numbers $L$ such that there exists a multiplicative Sidon set $A\subseteq\{1,2,\dots,n\}$ that intersects every interval $[x,x+L]\subseteq[1,n]$. Sárközy asked for estimates on $g(n)$, and he in particular asked whether one has $g(n)\le\sqrt n$ for every $n\in\mathbb{N}$. We first show that this estimate does indeed hold, with a proof that was autonomously discovered and formally verified in Lean by Aristotle. Next, we improve