We prove surjectivity of certain word maps on finite non-abelian simple groups. More precisely, we prove the following: if N is a product of two prime powers, then the word map (x, y) ↦ xNyN is surjective on every finite non-abelian simple group; if N is an odd integer, then the word map (x, y, z) ↦ xNyNzN is surjective on every finite quasisimple group. These generalize classical theorems of Burnside and Feit–Thompson. We also prove asymptotic results about the surjectivity of the word map (x, y) ↦ xNyN that depend on the number of prime factors of the integer N.
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