Binomial coefficients are (almost) never powers

Abstract

There is an epilogue to Bertrand’s postulate which leads to a beautiful result on binomial coefficients. In 1892 Sylvester strengthened Bertrand’s postulate in the following way:

$$In\;n > 2k,then\;at\;least\;one\;of\;the\;numbers\;n,n - 1,...,n - k + 1\;has\;a\;prime\;divisor\;p\;greater\;than\;k.$$

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