We give non-trivial upper bounds for the number of integral solutions, of given size, of a system of two quadratic form equations in five variables.

Suppose that ${\lambda }_1,{\lambda }_2,{\lambda }_3,{\lambda }_4$ are nonzero real numbers, not all negative, $\delta >0$, $𝒱$ is a well-spaced set, and the ratio ${\lambda }_1/{\lambda }_2$ is algebraic and irrational. Denote by $E(𝒱,N,\delta )$ the number of $v\in 𝒱$ with $v\le N$ such that the inequality $$|{\lambda }_1{p}_1^2+{\lambda }_2{p}_2^3+{\lambda }_3{p}_3^4+{\lambda }_4{p}_4^5-v|<v^{-\delta }$$ has no solution in primes $p_1$, $p_2$, $p_3$, $p_4$. We show that $$E(𝒱,N,\delta )\ll N^{1+2\delta -1/72+\epsilon }$$ for any $\epsilon >0$.

1. Introduction. A positive number which is a sum of two odd primes is called a Goldbach number. Let E(x) denote the number of even numbers not exceeding x which cannot be written as a sum of two odd primes. Then the Goldbach conjecture is equivalent to proving that E(x) = 2 for every x ≥ 4. E(x) is usually called the exceptional set of Goldbach numbers. In [8] H. L. Montgomery and R. C. Vaughan proved that $E\left(x\right)=O\left(x^{1-\Delta }\right)$ for some positive constant Δ > 0$.In\left[3\right]ChenandPanprovedthatonecantake\Delta >0.01.In\left[6\right],weprovedthatE\left(x\right)=O\left(x^{0.921}\right)$. In this paper we prove the following result. Theorem....

A new method for counting primes in a Beatty sequence is proposed, and it is shown that an asymptotic formula can be obtained for the number of such primes in a short interval.

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.