Sums of squares in $S$-integers

Sums of powers of consecutive q-integers

We give the q-analogue of the sums of the n-th powers of positive integers up to k-1.

将三个连续整数表示为三个立方数之和

Expressing three consecutive integers as sums of three cubes

本文研究将连续整数表示为三个立方数之和的问题，给出三个连续整数的若干参数化解，并讨论五至七个连续整数的平凡解，最后提出四个及以上连续整数的开放问题。

On the alternating sums of powers of consecutive q-integers

In this paper we construct $q$-Genocchi numbers and polynomials. By using these numbers and polynomials, we investigate the $q$-analogue of alternating sums of powers of consecutive integers due to Euler.

Representations of integers as sums of primes from a Beatty sequence

We study the problem of representing integers as sums of prime numbers from a fixed Beatty sequence $B_{α,β}$, where $α>1$ is irrational and of finite type.

Multiple harmonic sums and multiple harmonic star sums are (nearly) never integers

It is well known that the harmonic sum $H_n(1)=\sum_{k=1}^n\frac{1}{k}$ is never an integer for $n>1$. In 1946, Erdős and Niven proved that the nested multiple harmonic sum $H_n(\{1\}^r)=\sum_{1\le k_1<\dots<k_r\le n}\frac{1}{k_1\cdots k_r}$ can take integer values only for a finite number of positive integers $n$. In 2012, Chen and Tang refined this result by showing that $H_n(\{1\}^r)$ is an integer only for $(n,r)=(1,1)$ and $(n,r)=(3,2)$. In this paper, we consider the integrality problem for arbitrary multiple harmonic and multiple harmonic star sums and show that none of these sums is an…

A note on the alternating sums of powers of consecutive q-integers

In this paper we construct a new q-Euler numbers and polynomials. By using these numbers and polynomials, we give the interesting formulae related to alternating sums of powers of consecutive q-integers following an idea due to Euler.

On sums of S-integers of bounded norm

We prove an asymptotic formula for the number of S-integers in a number field K that can be represented by a sum of n S-integers of bounded norm.

Positive alternating sums of integer partitions

We combine an extended version of Bailey's transform with an identity of Bressoud and with some identities of Berkovich and Warnaar to prove a variety of positivity results for alternating sums involving partition functions.

Note on the sums of powers of consecutive $q$-integers

In this paper we construct the $q$-analogue of Barnes's Bernoulli numbers and polynomials of degree 2, for positive even integers, which is an answer to a part of Schlosser's question. For positive odd integers, Schlosser's question is still open. Finally, we will treat the $q$-analogue of the sums of powers of consecutive integers.

Integer ratios of consecutive alternating power sums

We give a characterization of all pairs $(k,n)$ of positive integers for which the ratio $$ \frac{1^k-2^k+3^k-\dots+(-1)^{n+1} n^k}{1^k-2^k+3^k-\dots+(-1)^{n}(n-1)^k} $$ of two consecutive alternating power sums is an integer.

Signed harmonic sums of integers with $k$ distinct prime factors

We give some theoretical and computational results on "random" harmonic sums with prime numbers, and more generally, for integers with a fixed number of prime factors.

On a conjecture for representations of integers as sums of squares and double shuffle relations

In this paper, we prove a conjecture of Chan and Chua for the number of representations of integers as sums of 8s integral squares. The proof uses a theorem of Imamoglu and Kohnen, and the double shuffle relations satisfied by the double Eisenstein series of level 2.

可表示为两个平方数之和的整数有限序列

Finite sequences of integers expressible as sums of two squares

本文构造了无穷多组满足特定间隔条件的整数序列，使其各项均可表示为两个非零整数平方和，并给出连续三项、四项、五项及公差为4的五项算术级数等参数化结果。

On Integer Sequences Associated To Two Distinct Sums

In this note, we show the existence of integer sequences of lengths at least 3 (except 7) such that for every integer in position $i\equiv 1\pmod{4}$ (respectively position $j\equiv 3\pmod{4}$), counting from left to right, the sum of the integer and the adjacent integer(s) has a constant sum $x$ (respectively $y$) with $x\ne y$.

Recursive formulas for sums of squares and sums of triangular numbers

We prove recursive formulas for sums of squares and sums of triangular numbers in terms of sums of divisors functions and we give a variety of consequences of these formulas. Intermediate applications include statements about positivity of the coefficients of some infinite products.

Sums of integers and sums of their squares

Suppose a positive integer $n$ is written as a sum of squares of $m$ integers. What can one say about the value $T$ of the sum of these $m$ integers itself? Which $T$ can be obtained if one considers all possible representations of $n$ as a sum of squares of $m$ integers? Denoting this set of all possible $T$ by $\mathscr{S}_m(n)$, Goldmakher and Pollack have given a simple characterization of $\mathscr{S}_4(n)$ using elementary arguments. Their result can be reinterpreted in terms of Mordell's theory of representations of binary integral quadratic forms as sums of squares of integral linear f…

探索大正整数表示为素数幂与整数幂之和：基于正密度子集的分析

Exploring the Representation of Large Positive Integers as Sums of Prime Powers and Integer Powers: Analysis with Positive Density Subsets

本文利用转换原理研究大正整数表示为素数幂与整数幂之和的问题，其中素数和整数幂分别取自正密度子集，拓展了加法数论中的表示理论。

Sumsets avoiding squarefree integers

We describe the structure of a set of integers $A$ of positive density $δ$, such that $A+A$ contains no squarefree integer. It turns out that the behaviour changes abruptly at the values $δ_0=1/4-\frac{2}{π^2}=0.0473...$ and $δ_1=1/18$.

A note on the sums of powers of consecutive q-integers

In this paper we construct the q-analogue of Barnes' Bernoulli numbers and plynomials of degree 2, which is an answer to a part of Schlosser's question. Finally, we treat the q-analogue of the sums of powers of consecutive integrs.

More on Faulhaber's theorem for sums of powers of integers

In this note we consider the theorem established in arXiv:1912.07171 concerning the sums of powers of the first $n$ positive integers, $S_k = 1^k + 2^k + \cdots + n^k$, and show that it can be used to demonstrate the classical theorem of Faulhaber for both cases of odd and even $k$.

Sums of Consecutive Prime Squares

We prove explicit bounds for the number of sums of consecutive prime squares below a given magnitude.

On integers which are representable as sums of large squares

We prove that the greatest positive integer that is not expressible as a linear combination with integer coefficients of elements of the set $\{n^2,(n+1)^2,\ldots \}$ is asymptotically $O(n^2)$, verifying thus a conjecture of Dutch and Rickett. Furthermore we ask a question on the representation of integers as sum of four large squares.

表示为模2、模3的较低次幂嵌套和的整数幂

Integer Powers Expressed as Nested Sums of Lower Powers modulo 2, 3

本文探索将正整数幂表示为较小正整数幂嵌套和的恒等式，建立了连续递减幂的基础恒等式并以此证明 Fermat 小定理，随后推导并严格证明了模2和模3的整数幂表示恒等式。

除固定整数外的整数平方和

Sums of squares of integers except for a fixed one

本文研究除固定整数外的整数平方和问题，对任意非负整数$n$找到了能表示所有相关正整数的最小平方数数量，推广了Dubouis在$n=0$时的结果。

The Sums of Two Squares do not have Metric Poissonian Pair Correlation

We prove that the sequence of the sums of two squares do not have metric Poissonian pair correlation.

Conjectures of Sun about sums of polygonal numbers

In this paper, we show that certain sums of generalized $m$-gonal numbers represent every positive integer if and only if they represent every positive integer up to an explicit bound $C_m$, verifying a conjecture of Sun for sufficiently large positive integers.

Congruences for sums of powers of an integer

For coprime positive integers $q$ and $e$, let $m(q,e)$ denote the least positive integer $t$ such that there exists a sum of $t$ powers of $q$ which is divisible by $e$. We prove an upper bound for $m(q.e)$ and investigate the case where $m(q,e)$ is "large". We also pay special attention to the situation where $e$ is a prime power.

Sums of Squares Bijective Parameter Representation

We give a bijective parameter representation for a sum of squares of numbers being equal to another sum of squares of numbers.

Sums over Vanishing Determinants

We study sums of arithmetic functions, defined on Gaussian integers and taken over those pairs of integers whose coordinates give rise to a singular system.

On Bourgain's bound for short exponential sums and squarefree numbers

We use Bourgain's recent bound for short exponential sums to prove certain independence results related to the distribution of squarefree numbers in arithmetic progressions.

On certain sums concerning the gcd's and lcm's of $k$ positive integers

We use elementary arguments to prove results on the order of magnitude of certain sums concerning the gcd's and lcm's of $k$ positive integers, where $k\ge 2$ is fixed. We refine and generalize an asymptotic formula of Bordellès (2007), and extend certain related results of Hilberdink and Tóth (2016). We also formulate some conjectures and open problems.

Perfect powers that are sums of consecutive squares

We determine all perfect powers that can be written as the sum of at most 10 consecutive squares.

A sum of three nonunit squares of integers

We say a positive integer is a sum of three nonunit squares if it is a sum of three squares of integers other than one. In this article, we find all integers which are sums of three nonunit squares assuming that the Generalized Riemann Hypothesis(GRH) holds. As applications, we find all integers, under the GRH only when $k=3$, which are sums of $k$ nonzero triangular numbers, sums of $k$ nonzero generalized pentagonal numbers, and sums of $k$ nonzero generalized octagonal numbers, respectively for any integer $k\ge 3$.

Restricted sums of four integral squares

We give a simple quaternionic proof of a recent result of Goldmakher and Pollack on restricted sums of four integral squares.

Sums of three squareful numbers

We investigate the frequency of positive squareful numbers x,y,z for which x+y=z.

有理整数作为单位之和——二次情况

Rational integers as sums of units -- the quadratic case

本文研究了二次数域中自然数作为k个单位和的渐进行为，解决了Jarden和Narkiewicz在2007年提出的问题。

Sums of powers of integers and hyperharmonic numbers

In this paper, we derive a formula for the sums of powers of the first $n$ positive integers, $S_k(n)$, that involves the hyperharmonic numbers and the Stirling numbers of the second kind. Then, using an explicit representation for the hyperharmonic numbers, we generalize this formula to the sums of powers of an arbitrary arithmetic progression. Moreover, we express the Bernoulli polynomials in terms of hyperharmonic polynomials and Stirling numbers of the second kind. Finally, we extend the obtained formula for $S_k(n)$ to negative values of $n$.

整数作为四个多边形数之和的表示与部分theta函数

Representations of integers as sums of four polygonal numbers and partial theta functions

研究整数作为最多四个不同m边形数之和的表示，证明非负参数下的表示数与任意整数参数下的表示数在渐进行为上相同。

Euler's and the Taxi Cab relations and other numbers that can be written twice as sums of two cubed integers

We show that Euler's relation and the Taxi-Cab relation are both solutions of the same equation. General solutions of sums of two consecutive cubes equaling the sum of two other cubes are calculated. There is an infinite number of relations to be found among the sums of two consecutive cubes and the sum of two other cubes, in the form of two families. Their recursive and parametric equations are calculated.

包含固定数量整数 pairwise 和的集合的基数

The cardinality of a set containing the pairwise sums of a fixed number of integers

研究显示，当集合A的大小超过n+1.2×10⁸时，存在五个不同整数的pairwise和均在A中。通过调整常数，可保证三个或四个整数的pairwise和也在A中。

Binary forms as sums of two squares and Châtelet surfaces

The representation of integral binary forms as sums of two squares is discussed and applied to establish the Manin conjecture for certain Châtelet surfaces defined over the rationals.

On supercongruences for truncated sums of squares of basic hypergeometric series

Congruences of truncated sums of infinite series do not directly extend to congruences of the truncated sums of higher powers of these infinite series. Guo and Zudilin recently established a variety of supercongruences for truncated sums of certain basic hypergeometric series. In this note we extend some of these supercongruences to the truncated sums of the squares of the corresponding series.

Integers that are sums of two rational sixth powers

We prove that $164634913$ is the smallest positive integer that is a sum of two rational sixth powers but not a sum of two integer sixth powers. If $C_{k}$ is the curve $x^{6} + y^{6} = k$, we use the existence of morphisms from $C_{k}$ to elliptic curves, together with the Mordell-Weil sieve, to rule out the existence of rational points on $C_{k}$ for various $k$.

整数表示为不同三的幂之和的商

Representations of integers as quotients of sums of distinct powers of three

本文研究哪些整数可表示为不同三的幂之和的商，提出必要且几乎充分的条件和生成所有表示的算法，并基于0,1-多项式对表示进行分类，完整描述了不超过364的整数及若干无限族。

Lucas-Carmichael 整数与基-$p$ 位数和的一个新刻画

A new characterization for the Lucas-Carmichael Integers and sums of base-$p$ digits

本文通过基-$p$ 位数和给出 Lucas-Carmichael 整数的充分必要刻画，研究其相关性质，并在素数 $k$-元组猜想下证明其无穷性。

Uniform Distribution of Hardy Sums

We employ the spectral theory of Eisenstein series to prove that the Hardy sums, integer-valued analogs of the classical Dedekind sums, are uniformly distributed in $\mathbf{Z} / m \mathbf{Z}$ for any integer $m > 1$.

Congruence conditions on the number of terms in sums of consecutive squared integers equal to squared integers

Considering the problem of finding all the integer solutions of the sum of $M$ consecutive integer squares starting at $a^{2}$ being equal to a squared integer $s^{2}$, it is shown that this problem has no solutions if $M\equiv3,5,6,7,8$ or $10 (mod\,12)$ and has integer solutions if $M\equiv0,9,24$ or $33 (mod\,72)$; or $M\equiv1,2$ or $16 (mod\,24)$; or $M\equiv11 (mod\,12)$. All the allowed values of $M$ are characterized using necessary conditions. If $M$ is a square itself, then $M\equiv1 (mod\,24)$ and $(M-1)/24$ are all pentagonal numbers, except the first two.

Multiplicative functions supported on the $k$-free integers with small partial sums

We provide examples of multiplicative functions $f$ supported on the $k$-free integers such that at primes $f(p)=\pm 1$ and such that the partial sums of $f$ up to $x$ are $o(x^{1/k})$. Further, if we assume the Generalized Riemann Hypothesis, then we can improve the exponent $1/k$: There are examples such that the partial sums up to $x$ are $o(x^{1/(k+\frac{1}{2})+ε})$, for all $ε>0$. This generalizes to the $k$-free integers the results of Aymone, `` A note on multiplicative functions resembling the {M}öbius function'', J. Number Theory, 212 (2020), pp. 113--121.

General solutions of sums of consecutive cubed integers equal to squared integers

All integer solutions $\left(M,a,c\right)$ to the problem of the sums of $M$ consecutive cubed integers $\left(a+i\right)^{3}$ ($a>1$, $0\leq i\leq M-1$) equaling squared integers $c^{2}$ are found by decomposing the product of the difference and sum of the triangular numbers of $\left(a+M-1\right)$ and $\left(a-1\right)$ in the product of their greatest common divisor $g$ and remaining square factors $δ^{2}$ and $σ^{2}$, yielding $c=gδσ$. Further, the condition that $g$ must be integer for several particular and general cases yield generalized Pell equations whose solutions allow to find all …

On a Conjecture of Sun about sums of restricted squares

In this paper, we investigate sums of four squares of integers whose prime factorizations are restricted, making progress towards a conjecture of Sun that states that two of the integers may be restricted to the forms $2^a3^b$ and $2^c5^d$. We obtain an ineffective generalization of results of Gauss and Legendre on sums of three squares and an effective generalization of Lagrange's four-square theorem.

Alternating Euler sums at the negative integers

We study three special Dirichlet series, two of them alternating, related to the Riemann zeta function. These series are shown to have extensions to the entire complex plane and we find their values at the negative integers (or residues at poles). These values are given in terms of Bernoulli and Euler numbers.

On squares of sums of three cubes

We show that almost every positive integer can be expressed as a sum of four squares of integers represented as the sums of three positive cubes.

Sums of products involving power sums of $φ(n)$ integers

A sequence of rational numbers as a generalization of the sequence of Bernoulli numbers is introduced. Sums of products involving the terms of this generalized sequence are then obtained using an application of the Faà di Bruno's formula. These sums of products are analogous to the higher order Bernoulli numbers and are used to develop the closed form expressions for the sums of products involving the power sums $\displaystyle Ψ_k(x,n):=\sum_{d|n}μ(d)d^k S_k(\frac{x}{d}), n\in\mathbb{Z}^+$ which are defined via the Möbius function $μ$ and the usual power sum $S_k(x)$ of a real or complex varia…

Sums of Two Squares Visualized

We provide a geometric interpretation of Brillhart's celebrated algorithm for expressing a prime $p\equiv 1\pmod 4$ as the sum of two squares.