Related papers: Exceptional sets in Waring's problem: two squares …

In this paper, it is established that every sufficiently large positive integer $n$ subject to $n\equiv0\pmod2$ can be represented as a sum of one square of prime and seventeen fifth powers of primes, which gives an enhancement upon the…

Every odd prime number p can be written in exactly (p + 1)/2 ways as a sum ab+cd of two ordered products ab and cd such that min(a, b) > max(c, d). An easy corollary is a proof of Fermat's Theorem expressing primes in 1 + 4N as sums of two…

Let $r_Q(n)$ be the representation number of a nonnegative integer $n$ by the quaternary quadratic form $Q=x_1^2+2x_2^2+x_3^2+x_4^2+x_1x_3+x_1x_4+x_2x_4$. We first prove the identity $r_Q(p^2n)=r_Q(p^2)r_Q(n)/r_Q(1)$ for any prime $p$…

Let $n>1$ be a positive integer. Let $R$ be a henselian local ring with residue field $k$ of $n$th level $s_n(k)$. We give some upper and lower bounds for the $n$th Waring number $w_n(R)$ in terms of $w_n(k)$ and $s_n(k)$. In large number…

We establish that, for almost all natural numbers $N$, there is a sum of two positive integral cubes lying in the interval $[N-N^{7/18+\epsilon},N]$. Here, the exponent $7/18$ lies half way between the trivial exponent $4/9$ stemming from…

Following Ekhad and Zeilberger (The Personal Journal of Shalosh B. Ekhad and Doron Zeilberger, Dec 5 2014; see also arXiv:1412.2035), we study the asymptotics for large $n$ of the number $A_{d,r}(n)$ of words of length $rn$ having $r$…

An asymptotic expansion for the generalised quadratic Gauss sum $$S_N(x,\theta)=\sum_{j=1}^{N} \exp (\pi ixj^2+2\pi ij\theta),$$ where $x$, $\theta$ are real and $N$ is a positive integer, is obtained as $x\rightarrow 0$ and…

A result of Wright from 1937 shows that there are arbitrarily large natural numbers which cannot be represented as sums of $s$ $k$th powers of natural numbers which are constrained to lie within a narrow region. We show that the analogue of…

The tensor square conjecture states that for $n \geq 10$, there is an irreducible representation $V$ of the symmetric group $S_n$ such that $V \otimes V$ contains every irreducible representation of $S_n$. Our main result is that for large…

Employing Br\"udern's and Wooley's new complification method, we establish an asymptotic Hasse principle for the number of solutions to a system of r_3 cubic and r_2 quadratic diagonal forms, when the number of cubic equations is at least…

We study the supersaturation problems of oddtown and eventown. Given a family $\mathcal A$ of subsets of an $n$ element set, let $op(\mathcal A)$ denote the number of distinct pairs $A,B\in \mathcal A$ for which $|A \cap B|$ is odd. We show…

This paper studies the minimal conditions under which we can establish asymptotic formulae for Waring's problem and other additive problems that may be tackled by the circle method. We confirm in quantitative terms the well-known heuristic…

Let N(n, t) be the minimal number of points in a spherical t-design on the unit sphere S^n in R^{n+1}. For each n >= 3, we prove a new asymptotic upper bound N(n, t) <= C(n)t^{a_n}, where C(n) is a constant depending only on n, a_3 <= 4,…

In this article, we study a sum of squares of integers except for a fixed one. For any nonnegative integer $n$, we find the minimum number of squares of integers except for $n$ whose sums represent all positive integers that are represented…

Waring's classical problem deals with expressing every natural number as a sum of g(k) k-th powers. Recently there has been considerable interest in similar questions for nonabelian groups, and simple groups in particular. Here the k-th…

We prove an upper bound for the number of representations of a positive integer $N$ as the sum of four $k$-th powers of integers of size at most $B$, using a new version of the Determinant method developed by Heath-Brown, along with recent…

New families of nonnegative biquadratic forms that have 8, 9 or 10 real zeros in $\mathbb{P}^2\times \mathbb{P}^2$ are constructed. These are the first examples with 8, 9 or 10 real zeros. It is known that nonnegative biquadratic forms with…

Waring problem for forms is important and classical in mathematics. It has been widely investigated because of its wide applications in several areas. In this paper, we consider the Waring problem for binary forms with complex coefficients.…

Let A be a subset of Z / NZ, and let R be the set of large Fourier coefficients of A. Properties of R have been studied in works of M.-C. Chang and B. Green. Our result is the following : the number of quadruples (r_1, r_2, r_3, r_4) \in…

For each odd prime power q, and each integer k, we determine the sum of the k-th powers of all elements x in F_q for which both x and x+1 are squares in F_q^*. We also solve the analogous problem when one or both of x and x+1 is a…