When $k\geqslant 4$ and $0\leqslant d\leqslant (k-2)/4$, we consider the system of Diophantine equations

\begin{align*}x_1^j+\ldots +x_k^j=y_1^j+\ldots +y_k^j\quad (1\leqslant j\leqslant k,\, j\ne k-d).\end{align*}

$d=o\!\left(k^{1/4}\right)$

In this paper, we show that every pair of sufficiently large even integers can be represented as a pair of eight prime cubes and k powers of $2$. In particular, we prove that $k=335$ is admissible, which improves the previous result.

Let N be a sufficiently large integer. We prove that, with at most $O(N^{23/48+\varepsilon })$ exceptions, all even positive integers up to N can be represented in the form $p_1^2+p_2^3+p_3^3+p_4^3+p_5^3+p_6^4$ , where $p_1,p_2,p_3,p_4,p_5,p_6$ are prime numbers.

We show that the system of equations

$$ \begin{align*} \sum_{i=1}^{s} (x_{i}^{\,j}-y_{i}^{\,j}) = a_{j} \quad (1 \leqslant j \leqslant k) \end{align*} $$

has appreciably fewer solutions in the subcritical range $s < \tfrac 12k(k+1)$ than its homogeneous counterpart, provided that $a_{\ell } \neq 0$ for some $\ell \leqslant k-1$ . Our methods use Vinogradov’s mean value theorem in combination with a shifting argument.

We prove uniform bounds for the Petersson norm of the cuspidal part of the theta series. This gives an improved asymptotic formula for the number of representations by a quadratic form. As an application, we show that every integer $n \neq 0,4,7 \,(\textrm{mod}\ 8)$ is represented as $n= x_1^2 + x_2^2 + x_3^3$ for integers $x_1,x_2,x_3$ such that the product $x_1x_2x_3$ has at most 72 prime divisors.

We give an upper bound for the minimum s with the property that every sufficiently large integer can be represented as the sum of s positive kth powers of integers, each of which is represented as the sum of three positive cubes for the cases $2\leq k\leq 4.$

This paper is concerned with the function r3(n), the number of representations of n as the sum of at most three positive cubes,

$$r_3(n) = {\mathrm{card}}\{\mathbf m\in\mathbb Z^3: m_1^3+m_2^3+m_3^3=n, m_j\ge1\}.$$

$${\sum_{m=1}^nr_3(m),\,\sum_{m=1}^nr_3(m)^2}$$

$${\sum_{\substack{ n\le x\\ n\equiv a\,\mathrm{mod}\,q }} r_3(n).\}$$

We improve some results in our paper [A. Languasco and A. Zaccagnini, ‘Short intervals asymptotic formulae for binary problems with prime powers’, J. Théor. Nombres Bordeaux30 (2018) 609–635] about the asymptotic formulae in short intervals for the average number of representations of integers of the forms $n=p_{1}^{\ell _{1}}+p_{2}^{\ell _{2}}$ and $n=p^{\ell _{1}}+m^{\ell _{2}}$, where $\ell _{1},\ell _{2}\geq 2$ are fixed integers, $p,p_{1},p_{2}$ are prime numbers and $m$ is an integer. We also remark that the techniques here used let us prove that a suitable asymptotic formula for the average number of representations of integers $n=\sum _{i=1}^{s}p_{i}^{\ell }$, where $s$, $\ell$ are two integers such that $2\leq s\leq \ell -1$, $\ell \geq 3$ and $p_{i}$, $i=1,\ldots ,s$, are prime numbers, holds in short intervals.

Let ${{a}_{1}},\,.\,.\,.\,,\,{{a}_{9}}$ be non-zero integers and $n$ any integer. Suppose that ${{a}_{1}}\,+\,.\,.\,.\,+\,{{a}_{9}}\,\equiv \,n$$\left( \bmod \,2 \right)$ and $\left( {{a}_{i}},\,{{a}_{i}} \right)\,=\,1$ for $1\,\le \,i\,<\,j\le \,9$. In this paper we prove that

• (i) if ${{a}_{j}}$ are not all of the same sign, then the cubic equation ${{a}_{1}}p_{1}^{3}\,+\,.\,.\,.\,+\,{{a}_{9}}p_{9}^{3}\,=\,n$ has prime solutions satisfying ${{p}_{j}}\,\ll \,{{\left| n \right|}^{{1}/{3}\;}}\,+\,\max {{\left\{ \left| {{a}_{j}} \right| \right\}}^{8+\varepsilon }}$;

• (ii) if all ${{a}_{j}}$ are positive and $n\,\gg \,\max {{\left\{ \left| {{a}_{j}} \right| \right\}}^{25+\varepsilon }}$ , then ${{a}_{1}}p_{1}^{3}\,+\,.\,.\,.\,+\,{{a}_{j}}p_{9}^{3}\,=\,n$ is soluble in primes $Pj$.

These results improve our previous results with the bounds $\max {{\left\{ \left| {{a}_{j}} \right| \right\}}^{14+\varepsilon }}$ and $\max \,{{\left\{ \left| {{a}_{j}} \right| \right\}}^{43+\varepsilon }}$ in place of $\max {{\left\{ \left| {{a}_{j}} \right| \right\}}^{8+\varepsilon }}$ and $\max {{\left\{ \left| {{a}_{j}} \right| \right\}}^{25+\varepsilon }}$ above, respectively.

We improve recent results of Bourgain and Shparlinski to show that, for almost all primes $p$, there is a multiple $mp$ that can be written in binary as

$$\begin{eqnarray}mp=1+2^{m_{1}}+\cdots +2^{m_{k}},\quad 1\leq m_{1}<\cdots <m_{k},\end{eqnarray}$$

$k=6$

$p$

$A=\langle g\rangle \subset \mathbb{F}_{p}^{\ast }$

$g\in \{2,3,5\}$

$|A|\gg p/(\log p)^{3}$

$A\cdot A+A\cdot A$

$\mathbb{F}_{p}$

Let $E(N)$ denote the number of positive integers $n\leqslant N$, with $n\equiv 4\;(\text{mod}\;24)$, which cannot be represented as the sum of four squares of primes. We establish that $E(N)\ll N^{11/32}$, thus improving on an earlier result of Harman and the first author, where the exponent $7/20$ appears in place of $11/32$.

Let ${{a}_{1}},...,{{a}_{9}}$ be nonzero integers and $n$ any integer. Suppose that ${{a}_{1}}+\cdot \cdot \cdot +{{a}_{9}}\,\equiv \,n\,\left( \bmod \,2 \right)$ and $\left( {{a}_{i}},\,{{a}_{j}} \right)\,=\,1$ for $1\,\le \,i\,<\,j\,\le \,9$. In this paper we prove the following:

(i) If ${{a}_{j}}$ are not all of the same sign, then the cubic equation ${{a}_{1}}p_{1}^{3}\,+\cdot \cdot \cdot +\,{{a}_{9}}\,p_{9}^{3}\,=\,n$ has prime solutions satisfying ${{p}_{j}}\,\ll \,{{\left| n \right|}^{{1}/{3}\;}}\,+\,\max {{\left\{ \left| {{a}_{j}} \right| \right\}}^{14+\varepsilon }}$.

(ii) If all ${{a}_{j}}$ are positive and $n\,\gg \,\max {{\left\{ \left| {{a}_{j}} \right| \right\}}^{43+\varepsilon }}$, then ${{a}_{1}}p_{1}^{3}\,+\cdot \cdot \cdot +\,{{a}_{9}}\,p_{9}^{3}\,=\,n$ is solvable in primes ${{p}_{j}}$.

These results are an extension of the linear and quadratic relative problems.

Paucity is established for a system of diagonal diophantine equations, in which the degrees are the odd numbers in ascending order.

We establish that almost all natural numbers n are the sum of four cubes of positive integers, one of which is no larger than n5/36. The proof makes use of an estimate for a certain eighth moment of cubic exponential sums, restricted to minor arcs only, of independent interest.

We prove that almost all natural numbers satisfying certain necessary congruence conditions can be written as the sum of two cubes of primes and two cubes of P2-numbers, where, as usual, we call a natural number a P2-number when it is a prime or the product of two primes. From this result we also deduce that every sufficiently large integer can be written as the sum of eight cubes of P2-numbers.

For any $n \geq 3$, let $F \in \mathbb{Z}[X_0,\ldots,X_n]$ be a form of degree $d\geq 5$ that defines a non-singular hypersurface $X \subset \mathbb{P}^{n}$. The main result in this paper is a proof of the fact that the number $N(F;B)$ of $\mathbb{Q}$-rational points on $X$ which have height at most $B$ satisfies $N(F;B)=O_{d,\varepsilon,n}(B^{n-1+\varepsilon}),$ for any $\varepsilon >0$. The implied constant in this estimate depends at most upon $d$, $\varepsilon$ and $n$. New estimates are also obtained for the number of representations of a positive integer as the sum of three $d$th powers, and for the paucity of integer solutions to equal sums of like polynomials.

Let $f \in \Z[x]$ be a polynomial of degree d. The paucity of non-trivial positive integer solutions to the equation $f(x_1)+f(x_2)=f(x_{3})+f(x_{4})$ is established, provided that $d \geq 7$. Also the corresponding situation is investigated for equal sums of three like polynomials.

As an application of the vector sieve and uniform estimates on the Fourier coefficients of cusp forms of half-integral weight, it is shown that any sufficiently large number $n\equiv 3$ (mod 24) with $5 \nmid n$ is expressible as a sum of three squares of integers having at most 521 prime factors.

Estimates are established for the number of integers of size N, in intervals of size $N^\theta$, that fail to admit a representation as the sum of s cubes (s = 5, 6). Thereby it is shown that almost all such integers are represented in the proposed manner. When s = 5 one may take $\theta = 10/21$, and when s = 6 one may take any $\theta >17/63$. Similar such conclusions are also established for the related problem associated with the expected asymptotic formula.

We investigate exceptional sets associated with various additive problems involving sums of cubes. By developing a method wherein an exponential sum over the set of exceptions is employed explicitly within the Hardy-Littlewood method, we are better able to exploit excess variables. By way of illustration, we show that the number of odd integers not divisible by 9, and not exceeding $X$, that fail to have a representation as the sum of 7 cubes of prime numbers, is $O\left( {{X}^{23/36+\varepsilon }} \right)$. For sums of eight cubes of prime numbers, the corresponding number of exceptional integers is $O\left( {{X}^{11/36+\varepsilon }} \right)$.