Related papers: Waring-Goldbach problems for one square and higher…

In this paper, we obtain formulas for the number of representations of positive integers as sums of arbitrarily many squares (and other polygonal numbers) with a certain natural weighting. The resulting weighted sums give Fourier…

We observe that the classical Faulhaber's theorem on sums of odd powers also holds for an arbitrary arithmetic progression, namely, the odd power sums of any arithmetic progression $a+b, a+2b, ..., a+nb$ is a polynomial in $na+n(n+1)b/2$.…

We study which integers are admissible as Fourier coefficients of even integer weight newforms. In the specific case of the tau-function, we show that for all odd primes $\ell < 100$ and all integers $m \geq 1$, we have $$ \tau(n) \neq \pm…

Recent results about sums of cubes of Fibonacci numbers [Frontczak, 2018] are extended to arbitrary powers.

For every even integer N, denote by D_{1,2}(N) the number of representations of N as a sum of a prime and an integer having at most two prime factors. In this paper, we give a new lower bound for D_{1,2}(N).

Using the fact that the number of combinations $p_{1}$, $p_{2}$, where $p_{1}$ and $p_{2}$ are odd primes, with $p_{1} \leq p_{2}$ and $p_{1} + p_{2} \leq 2N$ is equal to the total number of Goldbach pairs for all the even integers from 6…

We generalize two well-known enumerative facts. The first, due to Clebsch, says that a general binary sextic form is expressible as the sum of a cube and a square in 40 different ways. The second, due to Zariski and later Vakil, states that…

Positive polynomials arising from Muirhead's inequality, from classical power mean and elementary symmetric mean inequalities and from Minkowski's inequality can be rewritten as sums of squares.

In this Note, we try to study the relations between the Goldbach Conjecture and the least prime number in an arithmetic progression. We give a new weakened form of the Goldbach Conjecture. We prove that this weakened form and a weakened…

In this paper, we prove some results of restricted sums of four squares using arithmetic of quaternions in the ring of Lipschitz integers. For example, we show that every nonnegative integer $n$ can be written as $x^{2}+y^{2}+z^{2}+t^{2}$…

Let $A$ be a subset of primes up to $x$. If we assume $A$ is well-distributed (in the Siegel-Walfisz sense) in any arithmetic progressions to moduli $q\leqslant(\log x)^c$ for any $c>0$, then the sumset $A+A$ has density 1/2 in the natural…

In this paper, we discuss sums of powers of the positive integers and compute both the exponential and ordinary generating functions for these sums. We express these generating functions in terms of exponential and geometric polynomials and…

Let $g$ be sufficiently large, $b\in\{0,\ldots,g-1\}$, and $\mathcal{S}_b$ be the set of integers with no digit equal to $b$ in their base $g$ expansion. We prove that every sufficiently large odd integer $N$ can be written as $p_1 + p_2 +…

Drawing inspiration from the work of Nathanson and Yamada we prove that every even integer larger than $\exp (\exp (32.7))$ can be written as the sum of a prime and the product of at most two primes.

Let $n$ be a positive integer, $p$ be an odd prime and integers $a,b \not= 0$ with $gcd(a,b)=1$, $p \nmid ab$, and $p|(a^n \pm b^n)$, we prove the identity $$\nu_p(a^n \pm b^n)-\nu_p(n)=\nu_p(a^{p-1}-b^{p-1}).$$ An unintended interesting…

In this note we study holomorphic integer graded vertex superalgebras. We prove that all such vertex superalgebras of central charge 8 and 16 are purely even. For the case of central charge 24 we prove that the weight-one Lie superalgebra…

The Goldbach conjecture states that every even number can be decomposed as the sum of two primes. Let $D(N)$ denote the number of such prime decompositions for an even $N$. It is known that $D(N)$ can be bounded above by $$ D(N) \leq C^*…

Let $p$ be a prime, and $N$ be a positive integer not divisible by $p$. Denote by ${\rm ord}_N(p)$ the multiplicative order of $p$ modulo $N$. Let $\mathbb{F}_q$ represent the finite field of order $q=p^{{\rm ord}_N(p)}$. For $a,…

The aim of the present article is to explore the possibilities of representing positive integers as sums of other positive integers and highlight certain fundamental connections between their multiplicative and additive properties. In…

A classical theorem of Paley asserts the existence of an infinite family of quadratic characters whose character sums become exceptionally large. In this paper, we establish an analogous result for characters of any fixed even order.…