Related papers: On Waring's problem: two squares and three biquadr…

We show that almost all natural numbers n not divisible by 4, and not congruent to 7 modulo 8, are represented as the sum of three squares, one of which is the square of an integer no larger than (log n)^{1+e} (any e>0). This answers a…

For $n \geq 3$, an asymptotic formula is derived for the number of representations of a sufficiently large natural number $N$ as a sum of $r = 2^n + 1$ summands, each of which is an $n$-th power of natural numbers $x_i$, $i = \overline{1,…

We investigate sums of mixed powers involving two squares, two cubes, and various higher powers, concentrating on situations inaccessible to the Hardy-Littlewood method.

Let $R_s(n)$ denote the number of representations of the positive number $n$ as the sum of two squares and $s$ biquadrates. When $s=3$ or $4$, it is established that the anticipated asymptotic formula for $R_s(n)$ holds for all $n\le X$…

In this paper, it is proved that every sufficiently large even integer can be represented as the sum of two squares of primes, two cubes of primes, two biquadrates of primes and 16 powers of 2. Furthermore, there are at least 5.313% odd…

We provide estimates for $s^{\rm th}$ moments of biquadratic smooth Weyl sums, when $10\le s\le 12$, by enhancing the second author's iterative method that delivers estimates beyond the classical convexity barrier. As a consequence, all…

Let $\mu_1, \ldots, \mu_s$ be real numbers, with $\mu_1$ irrational. We investigate sums of shifted $k$th powers $\mathfrak{F}(x_1, \ldots, x_s) = (x_1 - \mu_1)^k + \ldots + (x_s - \mu_s)^k$. For $k \ge 4$, we bound the number of variables…

We investigate the number of representations of a large positive integer as the sum of two squares, two positive integral cubes, and two sixth powers, showing that the anticipated asymptotic formula fails for at most O((log X)^3) positive…

It is shown that every sufficiently large even integer is a sum of two primes and exactly 13 powers of 2. Under the Generalized Rieman Hypothesis one can replace 13 by 7. Unlike previous work on this problem, the proof avoids numerical…

Chebyshev famously observed empirically that more often than not, there are more primes of the form $3 \bmod 4$ up to $x$ than of the form $1 \bmod 4$. This was confirmed theoretically much later by Rubinstein and Sarnak in a logarithmic…

We demonstrate that there are infinitely many integers that cannot be expressed as the sum of two squares of integers and up to two non-negative integer powers of 2.

Inspired by the recent work by Nadji, Ahmia and Ram\'irez, we examined the arithmetic properties of $\bar{B}_{l_1,l_2} (n)$, the number of overpartitions of n whose parts are neither divisible by $l_1$ nor divisible by $l_2$. In particular,…

The problem of finding the sum of a polynomial's values is considered. In particular, for any $n\geq 3$, the explicit formula for the sum of the $n$th powers of natural numbers $S_n=\sum_{x=1}^{m}x^{n}$ is proved:…

A natural number is a binary $k$'th power if its binary representation consists of $k$ consecutive identical blocks. We prove an analogue of Waring's theorem for sums of binary $k$'th powers. More precisely, we show that for each integer $k…

We prove that every sufficiently large odd integer can be expressed as a sum of one square and fourteen fifth powers, all of primes. In addition, we establish that every sufficiently large even integer can be written as a sum of one square,…

For any fixed $k\geq 2$, we prove that every sufficiently large integer can be expressed as the sum of a $k$th power of a prime and a number with at most $M(k)=6k$ prime factors. For sufficiently large $k$ we also show that one can take…

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…

We study divisibility properties of certain sums and alternating sums involving binomial coefficients and powers of integers. For example, we prove that for all positive integers $n_1,..., n_m$, $n_{m+1}=n_1$, and any nonnegative integer…

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…

In this paper we give necessary and sufficient trace conditions for an n by n matrix over any commutative and associative ring with unity to be a sum of k-th powers of matrices over that ring, where n,k are integers greater equal 2. We…