Egyptian fractions: Erdős, always
---------------------------------------
We have barely finished talking about Erdős when he appears again. Nearly twenty years after his death, he is still publishing! He does so through the conjectures he formulated and left behind and, of course, through his friends and disciples, who sometimes find proofs of them. One such problem, as always simple to state but a headache to solve, concerned Egyptian fractions: “Can every integer be written as a sum of Egyptian fractions (fractions with numerator 1) whose denominators are products of two distinct prime numbers?” Erdős and his colleague and friend Ronald Graham, of the University of San Diego in California, were so sure of their result that they announced a proof in Richard Guy’s book Unsolved problems in number theory, published in 1981. Yet it was not until December 2015 that a proof of this result for three distinct prime numbers appeared, when Graham and Steve Butler of the University of Iowa—who had been a student of Graham’s wife—submitted a preprint of a paper they co-authored... with Erdős, increasing their Erdős number by one. Butler searched for decompositions by computer, and Graham’s calculations pushed beyond the machine’s limits. Confident that they can do better, the two authors already say they are convinced that every integer can be written as a sum of Egyptian fractions whose denominators are products of w distinct prime numbers. In any case, for w = 2, they have already written 1 as a sum of 48 unit fractions whose denominators are products of two distinct prime numbers.
-
Einstein's equations: a breakthrough
----------------------------------
