Sums of k unit fractions

Erdõs and Straus conjectured that for any positive integer n ≥ 2 the equation 4/n = 1/x + 1/y + 1/x has a solution in positive integers z, y, and z. Let m > k ≥ 3 and E_m,k(N) =| {n ≤ N | m/n = 1/t₁ + ⋯ + 1/t_khas no solution with t_i ∈ ℕ} | . We show that parametric solutions can be used to find upper bounds on E_m,k(N) where the number of parameters increases exponentially with k. This enables us to prove E_m,k(N) ≪ N exp(-c_n,k(log N)^1-1/2k-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.