# Erdős is still publishing

Paul Erdős is still publishing. His newest paper, with Ron Graham and Steve Butler,shows that any natural number can be written as a sum $\sum_1^\ell 1/a_i$ with $a_1 < \ldots < a_\ell$, where each denominator is the product of three distinct primes. A footnote, on the obvious generalization to expressing rational numbers in such a form, reads:
“One of the authors believes that all rational numbers can be expressed in this form, another author has doubts that every rational number can be expressed in this form, and the third author, already having looked in The BOOK at the answer, remains silent on this issue.” For more of the context on this paper, see Siobhan Roberts writing for the Simons Foundation.

“The BOOK”, of course, refers to Erdős’ frequent claim that God (who he did not believe in) had a book in heaven that had the best proof of every theorem.  A sample of this book can be found in Proofs from THE BOOK by Martin Aigner and Günter M. Ziegler.