A singular K3 surface related to sums of consecutive cubes

q-analogues of the sums of powers of consecutive integers

Following an idea due to J. Bernoulli, we explore the q-analogue of the sums of powers of consecutive integers.

On Waring's problem in sums of three cubes

We investigate the asymptotic formula for the number of representations of a large positive integer as a sum of $k$-th powers of integers represented as the sums of three positive cubes, counted with multiplicities. We also obtain a lower bound for the number of representations when the sums of three cubes are counted without multiplicities.

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.

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.

Cube sums of form $3p$ and $3p^2$ II

Let $p\equiv 2,5\mod 9$ be a prime. We prove that both $3p$ and $3p^2$ are cube sums. We also establish some explicit Gross-Zagier formulae and investigate the 3 part full BSD conjecture of the related elliptic curves.

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.

Sums of Cubes in Quaternion Rings

We investigate a version of Waring's Problem over quaternion rings, focusing on cubes in quaternion rings with integer coefficients. We determine the global upper and lower bounds for the number of cubes necessary to represent all such quaternions.

Correlation estimates for sums of three cubes

We establish estimates for linear correlation sums involving sums of three positive integral cubes. Under appropriate conditions, the underlying methods permit us to establish the solubility of systems of homogeneous linear equations in sums of three positive cubes whenever these systems have more than twice as many variables as equations.

A note on the alternating sums of powers of consecutive integers

Following an idea due to Euler, we evaluate the alternating sums of powers of consrcutive integers.

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.

围绕具有素数平方因子的立方体的连续强力三元组的不存在性

Nonexistence of Consecutive Powerful Triplets Around Cubes with Prime-Square Factors

本文证明了围绕具有素数平方因子的立方体的连续强力三元组不存在，结合模算术、p-adic赋值、Thue方程和椭圆曲线理论，推进了Erdős-Mollin-Walsh猜想的研究。

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.

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.

Factored closed-form expressions for the sums of cubes of Fibonacci and Lucas numbers

We obtain explicit factored closed-form expressions for Fibonacci and Lucas sums of the form \mbox{$\sum_{k = 1}^n {F_{2rk}^3 }$} and \mbox{$\sum_{k = 1}^n {L_{2rk}^3 }$}, where $r$~and~$n$ are integers.

On Waring's problem in sums of three cubes for smaller powers

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 $k$-th powers of integers represented as the sum of three positive cubes for the cases $2\leq k\leq 4.$

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.

On perfect powers that are sums of cubes of a five term arithmetic progression

We prove that the equation $(x-2r)^3 + (x-r)^3 + x^3 + (x+r)^3 + (x+2r)^3= y^p$ only has solutions which satisfy $xy=0$ for $1\leq r\leq 10^6$ and $p\geq 5$ prime.

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.

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.

Higher-dimensional Dedekind sums and their bounds arising from the discrete diagonal of the n-cube

Higher-dimensional Dedekind sums are defined as a generalization of a recent 1-dimensional probability model of Dilcher and Girstmair to a d-dimensional cube. The analysis of the frequency distribution of marked lattice points leads to new formulae in certain special cases, and to new bounds for the classical Dedekind sums. Upper bounds for the generalized Dedekind sums are defined in terms of 1-dimensional moments. In the classical two-dimensional case, the ratio of these sums to their upper bounds are cosines of angles between certain vectors of n-dimensional cones, conjectured to have a lar…

Lemke Oliver and Soundararajan bias for consecutive sums of two squares

In a surprising recent work, Lemke Oliver and Soundararajan noticed how experimental data exhibits erratic distributions for consecutive pairs of primes in arithmetic progressions, and proposed a heuristic model based on the Hardy--Littlewood conjectures containing a large secondary term, which fits the data very well. In this paper, we study consecutive pairs of sums of squares in arithmetic progressions, and develop a similar heuristic model based on the Hardy--Littlewood conjecture for sums of squares, which also explain the biases in the experimental data. In the process, we prove several …

Cube sums of form $3p$ and $3p^2$

Let $p\equiv 2,5\mod 9$ be an odd prime. In this paper, we prove that at least one of $3p$ and $3p^2$ is a cube sum by constructing certain nontrivial Heegner points. We also establish the explicit Gross-Zagier formulae for these Heegner points and give variants of the Birch and Swinnerton-Dyer conjecture of the related elliptic curves.

A diophantine equation for sums of consecutive like powers

We show that the diophantine equation $n^\ell+(n+1)^\ell + ...+ (n+k)^\ell=(n+k+1)^\ell+ ...+ (n+2k)^\ell$ has no solutions in positive integers $k,n \ge 1$ for all $\ell \ge 3$.

On perfect powers that are sums of cubes of a three term arithmetic progression

Using only elementary arguments, Cassels and Uchiyama (independently) determined all squares that are sums of three consecutive cubes. Zhongfeng Zhang extended this result and determined all perfect powers that are sums of three consecutive cubes. Recently, the equation $(x-r)^k + x^k + (x+r)^k$ has been studied for $k=4$ by Zhongfeng Zhang and for $k=2$ by Koutsianas. In this paper, we complement the work of Cassels, Koutsianas and Zhang by considering the case when $k=3$ and showing that the equation $(x-r)^3+x^3+(x+r)^3=y^n$ with $n\geq 5$ a prime and $0 < r \leq 10^6$ only has trivial solu…

Remark on a Paper by Izadi and Baghalaghdam about Cubes and Fifth Powers Sums

In this paper, we refine the method introduced by Izadi and Baghalaghdam to search integer solutions to the Diophantine equation $X_1^5+X_2^5+X_3^5=Y_1^3+Y_2^3+Y_3^3$. We show that the Diophantine equation has infinitely many positive solutions.

Newer sums of three cubes

The search of solutions of the Diophantine equation $x^3 + y^3 + z^3 = k$ for $k<1000$ has been extended with bounds of $|x|$, $|y|$ and $|z|$ up to $10^{15}$. The first solution for $k=74$ is reported. This only leaves $k=33$ and $k=42$ for $k<100$ for which no solution has yet been found. A total of 966 new solutions were found.

三个立方数之和，II

Sums of three cubes, II

本文改进迭代方法，给出 $4\le s\le 8$ 时三次光滑 Weyl 和 $s$ 次矩的估计，并由此改进不超过 $X$ 且可表为三个自然数立方和之整数个数的下界。

Sums of cubes with shifts

Let $μ_1, \ldots, μ_s$ be real numbers, with $μ_1$ irrational. We investigate sums of shifted cubes $F(x_1,\ldots,x_s) = (x_1 - μ_1)^3 + \ldots + (x_s - μ_s)^3$. We show that if $η$ is real, $τ>0$ is sufficiently large, and $s \ge 9$, then there exist integers $x_1 > μ_1, \ldots, x_s > μ_s$ such that $|F(\mathbf{x})- τ| < η$. This is a real analogue to Waring's problem. We then prove a full density result of the same flavour for $s \ge 5$. For $s \ge 11$, we provide an asymptotic formula. If $s \ge 6$ then $F(\mathbf{Z}^s)$ is dense on the reals. Given nine variables, we can generalise this to…

On powers that are sums of consecutive like powers

Let $k \ge 2$ be even, and let $r$ be a non-zero integer. We show that for almost all $d \ge 2$ (in the sense of natural density), the equation $$ x^k+(x+r)^k+\cdots+(x+(d-1)r)^k=y^n, \qquad x,~y,~n \in \mathbb{Z}, \qquad n \ge 2, $$ has no solutions.

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 the Diophantine equation in the form that a sum of cubes equals a sum of quintics

We show how sums of some $5th$ powers can be written as sums of some cubics

关于素数加平方或立方之和的计算结果

Computational results on sums of a prime with squares or cubes

本文通过计算验证了素数加两个平方数或立方数的表示猜想，证明了特定范围内所有整数均可表示，并利用椭圆曲线整点方法实现。

两个平方数之和的连续序列的正密度

Positive density for consecutive runs of sums of two squares

本文研究了两个平方数之和的连续序列在等差数列中的分布，证明对于任意奇数无平方因子模数和简化同余类，存在正密度的此类连续序列，这是 Maynard 素数序列结果的类比。

Sums of four prime cubes in short intervals

We prove that a suitable asymptotic formula for the average number of representations of integers $n=p_{1}^{3}+p_{2}^{3}+p_{3}^{3}+p_{4}^{3}$, where $p_1,p_2,p_3,p_4$ are prime numbers, holds in intervals shorter than the the ones previously known.

循环除代数中极大序上的矩阵作为平方和与立方和

Matrices over maximal orders in cyclic division algebras over Q as sums of squares and cubes

本文研究有理数域上奇素数次循环除代数中极大序上的矩阵，证明在特定条件下，任意阶数大于等于2的矩阵可表示为平方和与立方和，推广了已有结果。

算术多边形与连续平方和

Arithmetic Polygons and Sums of Consecutive Squares

本文研究了算术多边形与连续平方和的关系，证明了对于奇数N≥3存在至少一个N边算术多边形，并展示了存在无限多个偶数边的算术多边形。

The distribution of consecutive prime biases and sums of sawtooth random variables

In recent work, we considered the frequencies of patterns of consecutive primes $\pmod{q}$ and numerically found biases toward certain patterns and against others. We made a conjecture explaining these biases, the dominant factor in which permits an easy description but fails to distinguish many patterns that have seemingly very different frequencies. There was a secondary factor in our conjecture accounting for this additional variation, but it was given only by a complicated expression whose distribution was not easily understood. Here, we study this term, which proves to be connected to bot…

A Short Proof of a Known Relation for Consecutive Power Sums

We give a new short proof of the most simple relation between consecutive power sums of the first m positive integers.

能表示为两个立方和的整数

Integers that are sums of two cubes in the cyclotomic $\mathbb{Z}_p$-extension

研究能表示为两个立方和的整数在环论Z_p扩张中的性质及应用。

$\mathbb{F}_q[t]$ 中三个立方数的最佳和

Optimal sums of three cubes in $\mathbb{F}_q[t]$

本文使用圆法证明，在特定假设下 $\mathbb{F}_q[t]$ 中密度为 1 的元素可表示为三个本质最小次数立方数之和，将数量级结果提升为完整渐近公式。

Forbidden integer ratios of consecutive power sums

Let $S_k(m):=1^k+2^k+\cdots+(m-1)^k$ denote a power sum. In 2011 Bernd Kellner formulated the conjecture that for $m\ge 4$ the ratio $S_k(m+1)/S_k(m)$ of two consecutive power sums is never an integer. We will develop some techniques that allow one to exclude many integers $ρ$ as a ratio and combine them to exclude the integers $3\le ρ\le 1501$ and, assuming a conjecture on irregular primes to be true, a set of density $1$ of ratios $ρ$. To exclude a ratio $ρ$ one has to show that the Erdős-Moser type equation $(ρ-1)S_k(m)=m^k$ has no non-trivial solutions.

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.

On equal values of products and power sums of consecutive elements in an arithmetic progression

In this paper we study the Diophantine equation \begin{align*} b^k + \left(a+b\right)^k + &\left(2a+b\right)^k + \ldots + \left(a\left(x-1\right) + b\right)^k = \\ &y\left(y+c\right) \left(y+2c\right) \ldots \left(y+ \left(\ell-1\right)c\right), \end{align*} where $a,b,c,k,\ell$ are given integers under natural conditions. We prove some effective results for special values for $c,k$ and $\ell$ and obtain a general ineffective result based on Bilu-Tichy method.

On perfect powers that are sums of cubes of a seven term arithmetic progression

We prove that the equation $(x-3r)^3+(x-2r)^3 + (x-r)^3 + x^3 + (x+r)^3 + (x+2r)^3+(x+3r)^3= y^p$ only has solutions which satisfy $xy=0$ for $1\leq r\leq 10^6$ and $p\geq 5$ prime. This article complements the work on the equations $(x-r)^3 + x^3 + (x+r)^3 = y^p$ and $(x-2r)^3 + (x-r)^3 + x^3 + (x+r)^3 + (x+2r)^3= y^p$ . The methodology in this paper makes use of the Primitive Divisor Theorem due to Bilu, Hanrot and Voutier for a complete resolution of the Diophantine equation.

有限域上连续平方的幂和，以及多项式不同值之和

On sums of powers of consecutive squares over finite fields, and sums of distinct values of polynomials

本文研究有限域中连续元素为平方或非平方时的幂和，并将其用于计算四次多项式 X^4+aX^2+b 的像集元素之和，解决了两个已有猜想。

Sums of powers over equally spaced Fibonacci numbers

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

Sums of two cubes as twisted perfect powers, revisited

In this paper, we sharpen earlier work of the first author, Luca and Mulholland, showing that the Diophantine equation $$ A^3+B^3 = q^αC^p, \, \, ABC \neq 0, \, \, \gcd (A,B) =1, $$ has, for "most" primes $q$ and suitably large prime exponents $p$, no solutions. We handle a number of (presumably infinite) families where no such conclusion was hitherto known. Through further application of certain {\it symplectic criteria}, we are able to make some conditional statements about still more values of $q$, a sample such result is that, for all but $O(\sqrt{x}/\log x)$ primes $q$ up to $x$, the equa…

Superelliptic equations arising from sums of consecutive powers

Using only elementary arguments, Cassels solved the Diophantine equation $(x-1)^3+x^3+(x+1)^3=z^2$ in integers $x$, $z$. The generalization $(x-1)^k+x^k+(x+1)^k=z^n$ (with $x$, $z$, $n$ integers and $n \ge 2$) was considered by Zhongfeng Zhang who solved it for $k=2$, $3$, $4$ using Frey-Hellegouarch curves and their Galois representations. In this paper, by employing some sophisticated refinements of this approach, we show that the only solution for $k=5$ is $x=z=0$, and that there are no solutions for $k=6$. The chief innovation we employ is a computational one, which enables us to avoid the…

Sums of Consecutive Integers

A decomposition of a natural number n is a sequence of consecutive natural numbers that sums to n. We construct a one-to-one correspondence between the odd factors of a natural number and its decompositions. We study the decompositions by their lengths and introduce the concept of length spectrum. Also, we use diagrams to illustrate the ideas behind our proofs.

Congruent conditions on the number of terms, on the ratio number of terms to first terms and on the difference of first terms for sums of consecutive squared integers equal to squared integers

Sums of $M$ consecutive squared integers $\left(a+i\right)^{2}$ equaling squared integers (for $a\geq1$, $0\leq i\leq M-1$) yield certain linear groupings of pairs $\left(a_{1},a_{2}\right)$ of $a$ values for successive same values of $M$ when these are linked by $a_{1}+a_{2}=μM+1$ with $μ=\left(η/δ\right)\in\mathbb{\mathbb{Q}}^{+}$. In this paper, congruent conditions on $M,η,δ$, and on the difference $\left(a_{2}-a_{1}\right)$ are demonstrated for these linear groupings to hold. It is found that $η\equiv1\left(mod\,2\right)$ and $δ\equiv0,1$ or $5\left(mod\,6\right)$, and if $δ\equiv0\left(m…

Beyond endoscopy for the Symmetric Cube L-function

This paper is a first attempt at getting information on a symmetric power representation of a $GL_2$ automorphic form via a trace formula that is beyond endoscopic techniques. In particular, we study the symmetric third power representation for all forms over $\Q$ adjoined the cube roots of unity. A key tool needed in the study is an identity relating cubic exponential sums to Kloosterman sums. This very same identity is crucial to the fundamental lemma of a trace formula comparison in work of Mao and Rallis.

Equal sums of two cubes of binary quadratic forms

We give a complete description of all solutions to the equation $f_1^3 + f_2^3 = f_3^3 + f_4^3$ for quadratic forms $f_j \in \mathbb C[x,y]$ and show how Ramanujan's example can be extended to three equal sums of pairs of cubes. We also give a complete census in counting the number of ways a sextic $p \in \mathbb C[x,y]$ can be written as a sum of two cubes. The extreme example is $p(x,y) = xy(x^4-y^4)$, which has six such representations.

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

The problem of finding all the integer solutions in $a$, $M$ and $s$ of sums of $M$ consecutive integer squares starting at $a^{2}\geq1$ equal to squared integers $s^{2}$, has no solutions if $M\equiv3,5,6,7,8$ or $10\left(mod\,12\right)$ and has integer solutions if $M\equiv0,9,24% or %33\left(mod\,72\right)$; or $M\equiv1,2$ or $16\left(mod\,24\right)$; or $M\equiv11\left(mod\,12\right)$. In this paper, additional congruence conditions are demonstrated on the allowed values of $M$ that yield solutions to the problem by using Beeckmans' eight necessary conditions, refining further the possibl…

多个数表示为三个素数的立方和

Many Representations as Sums of Three Prime Cubes

本文针对Erdős关于素数立方和表示数的猜想给出无条件证明，利用Hecke等分布定理完成k=3的情况，再结合Green–Tao–Ziegler定理得到k=4时的下界，论证无法推广到k=4的情况。