We prove in this article that almost all large integers have a representation as the sum of a cube, a biquadrate, ..., and a tenth power.

We study the sum τ of divisors of the quadratic form m₁² + m₂² + m₃². Let $S₃\left(X\right)={\sum }_{1\le m₁,m₂,m₃\le X}\tau (m₁²+m₂²+m₃²)$. We obtain the asymptotic formula S₃(X) = C₁X³logX + C₂X³ + O(X²log⁷X), where C₁,C₂ are two constants. This improves upon the error term $O_{\epsilon }\left(X^{8/3+\epsilon }\right)$ obtained by Guo and Zhai (2012).

For a large odd integer N and a positive integer r, define b = (b₁,b₂,b₃) and $(N,r)=b\in \mathbb{N}³:1\le b_j\le r,(b_j,r)=1andb₁+b₂+b₃\equiv N\left(modr\right).$It is known that    $(N,r)=r²{\prod }_{p|r}p|N((p-1)(p-2)/p²){\prod }_{p|r}p\nmid N((p²-3p+3)/p²)$. Let ε > 0 be arbitrary and $R=N^{1/8-\epsilon }$. We prove that for all positive integers r ≤ R, with at most $O\left(Rlo{g}^{-A}N\right)$ exceptions, the Diophantine equation ⎧N = p₁+p₂+p₃, ⎨ $p_j\equiv b_j\left(modr\right),$ j = 1,2,3,$$⎩ with prime variables is solvable whenever b ∈ (N,r), where A > 0 is arbitrary.

Let $f\left(𝐩\right)$ be an additive form of degree $k$ with $s$ prime variables $p_1,p_2,\cdots ,p_s$. Suppose that $f$ has real coefficients ${\lambda }_i$ with at least one ratio ${\lambda }_i/{\lambda }_j$ algebraic and irrational. If s is large enough then $f$ takes values close to almost all members of any well-spaced sequence. This complements earlier work of Brüdern, Cook and Perelli (linear forms) and Cook and Fox (quadratic forms). The result is based on Hua’s Lemma and, for $k\ge 6$, Heath-Brown’s improvement on Hua’s Lemma.