Mathematical Trio Advances Centuries-Old Number Theory Problem | Quanta Magazine

The work — the first-ever limit on how many whole numbers can be written as the sum of two cubed fractions — makes significant headway on “a recurring embarrassment for number theorists.”

**Hasnain says:**

“Proving the full conjecture — that exactly half of all integers are the sum of two cubes — will require eventually tackling the set of numbers that have more than one associated matrix. This set, which Bhargava calls “very hazy,” includes both numbers that are the sum of two cubes and ones that aren’t. Handling such numbers will require completely new ideas, he said.

For now, researchers are happy to have finally settled the question for a substantial proportion of whole numbers, and are eager to probe the techniques in the proof further. “It’s one of those beautiful things: You can explain the result very easily, but the tools are very, very much at the cutting edge of number theory,” Sarnak said.”