The smallest denominator not contained in a unit fraction decomposition of 1 with fixed length

Abstract Let v (k) be the smallest integer larger than 1 that does not occur among the denominators in any identity of the form 1 = 1 n 1 + ⋯ + 1 n k, align*1=1n₁++1nₖ, align* where 1 less than or equals n 1 less than midline horizontal ellipsis less than n Subscript k 1 ≤ n 1 ⋯ n k 1 n₁ nₖ are pairwise distinct integers. In their 1980 monograph, Erdős and Graham asked for quantitative estimates on the growth of v (k) and suggested the lower bound v left parenthesis k right parenthesis much greater than k factorial v (k) ≫ k ! v (k) k!. In this paper we give the first known improvement and show that there exists an absolute constant c greater than 0 c > 0 c0 such that the inequality v (k) ≥ e c k 2 align*v (k) e^c k²align* holds for all positive integers k.