Sums of k unit fractions

Erdos and Straus conjectured that for any positive integer n greater than or equal to 2 the equation 4/n = 1/x + 1/y + 1/z has a solution in positive integers x; y, and z. Let m >k greater than or equal to 3 and E-m,E-k(N) = \ {n less than or equal to N \ m/n = 1/t(1) + 1/t(k) has no solution with t(i) is an element of N} \. We show that parametric solutions can be used to find upper bounds on E-m,E-k(N) where the number of parameters increases exponentially with k. This enables us to prove E-m,E-k(N) << N exp (-c(m,k)(log N)1-1/2(k-1)-1) with c(m,k) > 0. This improves upon earlier work by Viola (1973) and Shen (1986), and is an "exponential generalization" of the work of Vaughan (1970), who considered the case k = 3.