Egyptian fractions: Erdős, still going ---------------------------------------
We have barely finished talking about Erdős, and here he is again. Some twenty years after his death, he is still publishing! He does so through the conjectures he put forward and left behind, and of course also through the friends and disciples who (sometimes) find proofs for them. One of these problems, 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 the products of two distinct prime numbers?" He and his colleague and friend Ronald Graham, of the University of San Diego in California, were so confident in their results that they announced a proof in Richard Guy's book Unsolved problems in number theory, published in 1981. And yet it was not until December 2015 that a proof of this result, for three distinct prime numbers, was finally submitted as a preprint by Graham and Steve Butler of the University of Iowa — a student of Graham's wife — in a paper they co-signed… with Erdős, just enough to raise their Erdős number by one. Butler searched for decompositions by computer, and Graham's calculations made it possible to push past the machine's limits. Confident that they could do even better, the two authors have already declared themselves convinced that every integer can be written as a sum of Egyptian fractions whose denominator is a product of w distinct prime numbers. In any case, for w = 2, they have already managed to write 1 as a sum of 48 unit fractions whose denominator is a product of two distinct prime numbers.
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Einstein's equations: an advance ----------------------------------