A famous conjecture of Erdős and Straus is that for every integer \(n\ge 2\) , 4/n can be represented as \(1/x+1/y+1/z\) , where x, y, z are positive integers. This conjecture was generalized to 5/n by Sierpiński, and then Schinzel conjectured that for every integer \(m\ge 4\) there is a bound \(n_m\) such that the fraction m/n is the sum of 3 unit fractions for all integers \(n\ge n_m\) . Leveraging and generalizing work of Elsholtz and Tao, we show that if \(n_m\) exists it must be at least \(\exp (m^{1/3+o(1)})\) ; that is, there are numbers n this large for which m/n is not the sum of 3 unit fractions. We prove a weaker, but numerically explicit version of this theorem, showing that for \(m\ge 6.52\times 10^9\) there is a prime \(p\in (m^2,2m^2)\) with m/p not the sum of 3 unit fractions, and report on some extensive numerical calculations that support this assertion with the much smaller bound \(m\ge 20\) . A result of Vaughan is that for each m, most n’s have m/n representable; we make the dependence on m in this result explicit. In addition, we prove a result generalizing the problem to the sum of j unit fractions.
