Euler Numbers: [GKP88] e(1) = 2 e(n) = e(1)*e(2)*...*e(n-1) + 1 for n>1. Problem: Kellogg [GKP Exc. 4.59], [IzK99] Prove that if x1, ..., xn are positive integers and a rational a with 1/x1 + 1/x2 + ... + 1/xn + 1/a = 1, and a >= max max(x1, x2, ..., xn) then a+1 <= e(n+1). References: Cur22: D. R. Curtiss; On Kellogg's Diophantine problem, AMM 29 (1922) 380-387 Erd50: P. Erd\"os; Az 1/x1 + 1/x2 + ... + 1/xn = a/b egyenlet egesz szamu megoldasairol, Matematikai Lapok 1 (1950) 192-209 [english abstract on page 210] - solves Kellogg's problem too. GKP88: Ronald L. Graham, Donald E. Knuth, Oren Pataschnik; Concrete Mathematics, Addison Wessley, Reading (1994) 2nd Ed. Chap 4: Number Theory - Exc: 4.59 (Kellogg Problem) IzK99: O. T. Izhboldn, L. D. Kurlyandchik; One's best approach, Quantum (March/April 1999) 24-26 - Kellogg Problem -- mailto:Torsten.Sillke@uni-bielefeld.de http://www.mathematik.uni-bielefeld.de/~sillke/