18 Binomial coefficients are (almost)never powers Now since mi≤n/e≤n/3 we finally obtain kn2/3 kmim3 (k-1)mim3 n, or k3>n.With this contradiction,the proof is complete. 口 References [1]P.ERDOS:A theorem of Sylvester and Schur.J.London Math.Soc.9 (1934). 282-288. [2]P.ERDOS:On a diophantine equation,J.London Math.Soc.26 (1951). 176-178. [3]J.J.SYLVESTER:On arithmetical series,Messenger of Math.21(1892),1-19, 87-120:Collected Mathematical Papers Vol.4,1912,687-731.18 Binomial coefficients are (almost) never powers Now since mi ≤ n1/ ≤ n1/3 we finally obtain kn2/3 ≥ km1m3 > (k − 1)m1m3 > n, or k3 > n. With this contradiction, the proof is complete. References [1] P. ERDOS˝ : A theorem of Sylvester and Schur, J. London Math. Soc. 9 (1934), 282-288. [2] P. ERDOS˝ : On a diophantine equation, J. London Math. Soc. 26 (1951), 176-178. [3] J. J. SYLVESTER: On arithmetical series, Messenger of Math. 21 (1892), 1-19, 87-120; Collected Mathematical Papers Vol. 4, 1912, 687-731.