Terence Tao, a mathematician at the University of California, Los Angeles, and a winner of the Fields Medal in 2006, submitted a paper1 to the arXiv preprint server on 17 September that claims to prove a number theory conjecture posed by mathematician Paul Erdős in the 1930s.

“Terry Tao just dropped a bomb,” tweeted Derrick Stolee, a mathematician at Iowa State University in Ames, the day the paper detailing the solution appeared online.

Terence Tao/CC 2.0 Paul Erdős and Terence Tao study maths at the University of Adelaide in Australia, in a photo taken when Tao was 10 years old.

Like many puzzles in number theory, the Erdős discrepancy problem is simple to state but devilishly difficult to prove. Erdős, who died in 1996, speculated that any infinite string of the numbers 1 and −1 could add up to an arbitrarily large (positive or negative) value by counting only the numbers at a fixed interval for a finite number of steps.

The task is intuitively easy for some arrangements — tallying digits at any interval in a sequence that is all 1s will add up to a big number. And in an alternating sequence of 1s and −1s, choosing every second digit will do the job. But Erdős conjectured that it was true for any such sequence.

Tao’s proof shows that Erdős was right: these sums can, in fact, grow infinitely large for any arbitrary sequence, although he did not provide a way to calculate their value for a given instance.

Although the proof has not undergone a rigorous peer review, experts expressed no concern over whether it would survive a critical look. “I’m completely confident,” says Gil Kalai, a mathematician at Hebrew University in Jerusalem, Israel, adding that he expects the review to take little time.

Chris Cesare

25 September 2015


Tao, T. Preprint at http://arxiv.org/abs/1509.05363 (2015).