Madison Gamble, Spencer Hamblen, Blake Schildhauer, Chung Truong
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.
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Lagrange's Four Squares Theorem states that any positive integer can be expressed as the sum of four integer squares. We investigate the analogous question over Quaternion rings, focusing on squares of elements of Quaternion rings with integer coefficients. We determine the minimum necessary number of squares for infinitely many Quaternion rings, and give global upper and lower bounds.
We use a representability theorem of G. L. Watson to examine sums of squares in Quaternion rings with integer coefficients. This allows us to determine a large family of such rings where every element expressible as the sum of squares can be written as the sum of 3 squares.
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。
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.
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.
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.
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.
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.
Euler noted the relation $6^3=3^3+4^3+5^3$ and asked for other instances of cubes that are sums of consecutive cubes. Similar problems have been studied by Cunningham, Catalan, Gennochi, Lucas, Pagliani, Cassels, Uchiyama, Stroeker and Zhongfeng Zhang. In particular Stroeker determined all squares that can be written as a sum of at most $50$ consecutive cubes. We generalize Stroeker's work by determining all perfect powers that are sums of at most $50$ consecutive cubes. Our methods include descent, linear forms in two logarithms, and Frey-Hellegouarch curves.
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.
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 $2\leq k\leq 4.$
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.
We construct vertex transitive lattices on products of trees of arbitrary dimension $d \geq 1$ based on quaternion algebras over global fields with exactly two ramified places. Starting from arithmetic examples, we find non-residually finite groups generalizing earlier results of Wise, Burger and Mozes to higher dimension. We make effective use of the combinatorial language of cubical sets and the doubling construction generalized to arbitrary dimension. Congruence subgroups of these quaternion lattices yield explicit cubical Ramanujan complexes, a higher dimensional cubical version of Ramanuj…
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…
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.
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…
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.
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.
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。
We study the surface arising from the diophantine equation $m^3+(m+1)^3+...+(m+k-1)^3=l^2$. It turns out that this is a $K3$ surface with Picard number 20. We stduy its aritmetic properties in detail. We construct elliptic fibrations on it, and we find a parametric solution to the original equation. Also, we determine the Hasse-Weil zeta function of the surface over $Q$.
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…
We show how sums of some $5th$ powers can be written as sums of some cubics
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。
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.
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。
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。
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。
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.
Recent results about sums of cubes of Fibonacci numbers [Frontczak, 2018] are extended to arbitrary powers.
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…
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.
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.
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。
Sums of rational cubes and the $3$-Selmer group
本研究得到j不变量为0的椭圆曲线三次扭族中3-塞尔默群的分布，改进了两有理立方和整数密度的上下界，通过强化解析核心、证明三线性大筛法解决了此前方法的挑战。
On sums of unequal powers of primes and powers of 2
本文证明了充分大的偶数可表示为素数平方、立方、四次方及2的幂之和，且至少5.313%的奇数可表示为素数平方、立方与四次方之和，改进了R. Zhang的结果。
Sums of three cubes over a function field
本文通过函数域版本的圆方法，在假设比率猜想和特征大于3的条件下，证明了函数域F_q[t]中正比例元素可表示为三次立方的最小度数和，并探讨了二次狄利克雷L函数的类似猜想。
Binary Cubic Forms and Rational Cube Sum Problem
本文利用整二元三次型研究有理数立方和问题，无条件证明了在特定剩余类中存在无穷多个素数可表示为两个有理数的立方和。
We solve the equation $f(x,y)^3 + g(x,y)^3 = x^3 + y^3$ for homogeneous $f, g \in \mathbb C(x,y)$, completing an investigation begun by Viète in 1591. The usual addition law for elliptic curves and composition give rise to two binary operations on the set of solutions. We show that a particular subset of the set of solutions is ring-isomorphic to $\mathbb Z[e^{2 πi / 3}]$.
A proof of the $4,7$ cases of Sylvester's conjecture on cube sums
本文利用椭圆曲线全 BSD 猜想和 Unbounded Denominators 猜想的最新进展，证明了模 9 余 4 或 7 的素数均可表示为两个有理数的立方和，解决了 Sylvester 猜想的三分之二。
Additive representation in short intervals, II: sums of two like powers
本文证明几乎所有自然数附近存在两个正整数的立方和，并将结果推广到更高次幂，改进了短区间中的加法表示。
On perfect powers that are sums of cubes of a nine term arithmetic progression
本文研究九项等差数列立方和等于完全方幂的方程，在特定条件下证明大素数指数时仅有平凡解，而对指数2和3通过椭圆曲线构造出无穷多解。
Let $g:\mathbb{N}\to\{-1,1\}$ be a completely multiplicative function, $μ$ be the Möbius function and $μ_2^2(n)$ be the indicator that $n$ is cubefree. We prove that $f=μ^2g$ and $f=μ_2^2g$ have unbounded partial sums. Our proofs are built upon Klurman and Mangerel's proof of Chudakov's conjecture, Klurman's work on correlations of multiplicative functions and Tao's resolution of the Erdős discrepancy problem.
In this paper we consider the problem of estimating character sums to composite modulus and obtain some progress towards removing the cubefree restriction in the Burgess bound. Our approach is to estimate high order moments of character sums in terms of solutions to congruences with Kloosterman fractions and we deal with this problem by extending some techniques of Bourgain, Garaev, Konyagin and Shparlinski and Bourgain and Garaev from the setting of prime modulus to composite modulus. As an application of our result we improve an estimate of Norton.
Sums of cubes and the Ratios Conjectures
本文在 Ratios 猜想等标准数论假设下消除了 Hooley-Heath-Brown 关于六个立方和方程解数上界中的 $\varepsilon$ 因子，进而证明几乎所有符合条件的整数是三个立方数之和，并在最强假设下得到幂节省渐近公式。
In this paper, we investigate a number of $q$-supercongruences on double and triple sums. By means of a lemma devised by El Bachraoui and its generalization, we transform some $q$-supercongruences on double and triple sums into the $q$-supercongruences of the square and cube of truncated basic hypergeometric series.Our main tools still involve the `creative microscoping' method introduced by Guo and Zudilin, a lemma designed by Guo and Li and the Chinese remainder theorem.
We investigate sums of mixed powers involving two squares, two cubes, and various higher powers, concentrating on situations inaccessible to the Hardy-Littlewood method.
We apply a method of Davenport to improve several estimates for slim exceptional sets associated with the asymptotic formula in Waring's problem. In particular, we show that the anticipated asymptotic formula in Waring's problem for sums of seven cubes holds for all but $O(N^{1/3+ε})$ of the natural numbers not exceeding $N$.
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 prove a discriminant criterion for every 2 by 2 matrix over an order R to be sums of cubes and fourth powers over R. We also show that if q is a prime and n greater equal 2, then every n by n matrix over the ring of integers O, of a quadratic number field is a sum of q-th powers (of matrices) over O if and only if q is coprime to the discriminant of the quadrati…
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