A ternary additive problem

Waring-Goldbach problem for unlike powers

“…+ε , Hoffman and Yu [5] showed that θ(k) grows exponentially, whereas, Theorem 1.3 implicates that θ(k) = k 2 2 + O(k) with polynomial growth. Finally, we consider the problem of representing a large odd integer n in the form…”

“…Obviously, one hasm (α)S 2 k (α) dα ≪ N −θ 4 (k)+ε . (6.2)It suffices to prove thatM(2K)\M(K) S 8 3 (α)S 2 k (α) dα ≪ Nwith N By Lemma 4.2 and Lemma 5.2 in[5], one hasM(2K)\M(K) −θ 4 (k)+ε , since 3 2k > θ 4 (k)for all k 4. This establishes (6.2).…”

Sums of One Prime Power and Four Prime Cubes in Short Intervals

“…Tis result has been improved by Bauer [2], Bauer [3], and Zhao [4], and the latest result is O(N 1− (1/16)+ϵ ). For general k ⩾ 5, the best result was given by Hofman and Yu [5] which is O(N 1− (47/420•2 s )+ϵ ) where s � [k + 1/2].…”