The asymptotic formula in Waring’s problem for one square and seventeen fifth powers

摘要

Let R _k,s (n) denote the number of solutions of the equation n = x2 + y₁k + y₂k + ¼+ y_sk{n= x^2 + y_1^k + y_2^k + cdots + y_s^k} in natural numbers x, y ₁, . . . , y _s . By a straightforward application of the circle method, an asymptotic formula for R _k,s (n) is obtained when k ≥ 3 and s ≥ 2 k–1 + 2. When k ≥ 6, work of Heath-Brown and Boklan is applied to establish the asymptotic formula under the milder constraint s ≥ 7 · 2 k–4 + 3. Although the principal conclusions provided by Heath-Brown and Boklan are not available for smaller values of k, some of the underlying ideas are still applicable for k = 5, and the main objective of this article is to establish an asymptotic formula for R _5,17(n) by this strategy.