Eventually greedy best Egyptian underapproximations of rational numbers via optimal control
A 75-year-old puzzle about breaking fractions into simpler pieces
Mathematicians have solved a problem posed by Erdős and Graham in the 1970s about ancient Egyptian fractions—a method of writing rational numbers as sums of unit fractions (fractions with numerator 1). The researchers proved that every positive rational number can be built using a 'greedy' approach that always picks the largest possible unit fraction at each step, whether or not you're allowed to repeat the same denominator.
Egyptian fractions aren't just historical curiosities—they appear in computer algorithms, cryptography, and number theory. This proof settles a foundational question about whether the simplest approximation method always works, which establishes new bounds on how large denominators can grow when repeatedly breaking down fractions. The work also introduces an optimal control framework that other researchers can now apply to similar problems in discrete mathematics.