A 27-year-old doctoral student at Princeton has solved one of the longest-standing open problems in number theory using an approach so elementary that experts are still coming to terms with having missed it.
In 1964, Paul Erdős — mathematics' most prolific problem-poser — conjectured that within any sufficiently large set of integers, you can always find a subset whose elements sum to zero. The precise conditions under which this is guaranteed have eluded mathematicians for six decades. In March 2026, Priya Anand, a third-year PhD student at Princeton, posted a 28-page preprint that the mathematical community has now verified as a complete proof.
The Erdős conjecture in its simplest form states that given any set of n integers where n is sufficiently large relative to the maximum element size, there must exist a non-empty subset that sums to zero. The challenge has been defining exactly when "sufficiently large" kicks in and proving that no exceptions exist. For 60 years, mathematicians found partial results, counterexamples in edge cases, and bounds that almost worked — but never a complete proof.
What stunned the mathematical community was not just the proof's correctness but its accessibility. Anand's approach uses a clever reframing of the problem as a question about linear independence over finite fields — a perspective available to undergraduate mathematics students. "It's the kind of proof that makes you feel foolish for not seeing it yourself," wrote one Fields Medallist in a widely circulated review. "And that is the hallmark of a great mathematical insight."
"I didn't think I was solving a 60-year problem. I thought I was working on a homework exercise I had misunderstood."
— Priya Anand, Princeton University, 2026