Mathematicians long wondered whether it's possible to express the number 33 as the sum of three cubes--that is, whether the equation 33 x³ y³ z³ has a solution. They knew that 29 could be written as 3³ 1³ 1³, for instance, whereas 32 is not expressible as the sum of three integers each raised to the third power. But the case of 33 went unsolved for 64 years. Booker found this odd trio of 16-digit integers by devising a new search algorithm to sift them out of quadrillions of possibilities. The algorithm ran on a university supercomputer for three weeks straight.

In a report posted online last week, Peter Scholze of the University of Bonn and Jakob Stix of Goethe University Frankfurt describe what Stix calls a "serious, unfixable gap" within a mammoth series of papers by Shinichi Mochizuki, a mathematician at Kyoto University who is renowned for his brilliance. Posted online in 2012, Mochizuki's papers supposedly prove the abc conjecture, one of the most far-reaching problems in number theory. Original story reprinted with permission from Quanta Magazine, an editorially independent publication of the Simons Foundation whose mission is to enhance public understanding of science by covering research developments and trends in mathematics and the physical and life sciences. Despite multiple conferences dedicated to explicating Mochizuki's proof, number theorists have struggled to come to grips with its underlying ideas. His series of papers, which total more than 500 pages, are written in an impenetrable style, and refer back to a further 500 pages or so of previous work by Mochizuki, creating what one mathematician, Brian Conrad of Stanford University, has called "a sense of infinite regress."

At the Hotel Parco dei Principi in Rome, in September of 1973, the Hungarian mathematician Paul Erd?s approached his friend Richard Guy with a request. He said, "Guy, veel you have a coffee?" It cost a dollar, a small fortune to a professor of mathematics at the hinterland University of Calgary who was not much of a coffee drinker. Yet, as Guy later recalled--during a memorial talk following Erd?s's death at age 83 two decades ago--he was curious why the great man had sought him out. Guy and Erd?s were in the Eternal City for an international colloquium on combinatorial theory, so Erd?s--who sustained himself with espresso and other stimulants, worked on math problems 19 hours a day, and in his lifetime published in excess of 1,500 papers with more than 500 collaborators--most likely had another problem on the go.