Big Advance on Simple-Sounding Math Problem Was a Century in the Making
An editorially independent publication supported by the Simons Foundation.
Addition and multiplication are each relatively straightforward operations. But when the two are combined, they lead to deep questions that mathematicians are still working to understand.
One morning last November, the mathematician Hector Pasten finally solved the problem that had been dogging him for more than a decade by using a time-tested productivity hack: procrastination.
He was supposed to be writing a final exam for his number theory class at the Pontifical Catholic University of Chile in Santiago. To avoid the task, he started pondering, for the umpteenth time, one of his favorite sequences: 2, 5, 10, 17, 26 and so on, the list of all numbers of the form n2 + 1 (where n is a whole number).
Mathematicians have used this sequence for over a century to probe the fraught relationship between addition and multiplication, a tension that lies at the heart of number theory. Fundamental problems about multiplication — about, say, how numbers factor into primes — suddenly become much deeper and more challenging as soon as addition enters the picture. One of math’s biggest open questions, for example, asks whether every even number larger than 2 is the sum of two primes; another asks whether there are infinitely many pairs of primes that differ by only 2, such as 11 and 13.
