Pranabesh Das, Pallab Kanti Dey, Angelos Koutsianas, Nikos Tzanakis
In this paper we determine the perfect powers that are sums of three fifth powers in an arithmetic progression. More precisely, we completely solve the Diophantine equation $$ (x-d)^5 + x^5 + (x + d)^5 = z^n,~n\geq 2, $$ where $d,x,z \in \mathbb{Z}$ and $d = 2^a5^b$ with $a,b\geq 0$.
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We show that the generalized Fermat equations with signatures (5,5,7), (5,5,19), and (7,7,5) (and unit coefficients) have no non-trivial primitive integer solutions. Assuming GRH, we also prove the nonexistence of non-trivial primitive integer solutions for the signatures (5,5,11), (5,5,13), and (7,7,11). The main ingredients for obtaining our results are descent techniques, the method of Chabauty-Coleman, and the modular approach to Diophantine equations.
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.
On the sum of fifth powers in arithmetic progression
本文针对互素条件下等差数列五次幂和等于 $p$ 次幂的 Diophantine 方程，结合模方法、Frey 曲线、模形式与辛论证，给出不可解性证明方案，并处理 $d=2,3$ 及特定模 $9$ 渐近情形。
On Waring-Goldbach problem for one square and seventeen fifth powers of primes
本文证明每个充分大的偶数可表示为一个素数平方与十七个素数五次幂之和，改进了Brüdern和Kawada的先前结果。
The Minimal Absolute Value of Sums of Fifth Roots of Unity
本文确定了可重复选取的n个五次单位根的非零和的最小绝对值，并刻画了达到最小值的和的结构。
Waring-Goldbach problems for one square and higher powers
本文研究了用一个平方和更高次幂表示足够大的整数的问题，证明了奇数和偶数均可表示为素数的平方、四次幂和五次幂的组合。
Zagier provided eleven conjectural rank two examples for Nahm's problem. All of them have been proved in the literature except for the fifth example, and there is no $q$-series proof for the tenth example. We prove that the fifth and the tenth examples are in fact equivalent. Then we give a $q$-series proof for the fifth example, which confirms a recent conjecture of Wang. This also serves as the first $q$-series proof for the tenth example, whose explicit form was conjectured by Vlasenko and Zwegers in 2011.
We determine all perfect powers that can be written as the sum of at most 10 consecutive squares.
We present lower bounds on the sum and product of the distinct prime factors of an odd perfect number, which provide a lower bound on the size of the odd perfect number as a function of the number of its distinct prime factors.
The aim of this note is to show that any even perfect number, other than $6$, can be written as the sum of 5 cubes of natural numbers. We also conjecture that any even perfect number, other than $6$, can be written as the sum of only 3 cubes of natural numbers.
Matrices over non-commutative rings as sums of fifth and seventh powers
本文证明在非交换环上，当 $n<p$ 时，矩阵为 $p$ 次幂之和当且仅当其迹如此，推广了已有结果至 $p=5,7$。
In this paper, we consider the problem about finding out perfect powers in an alternating sum of consecutive cubes. More precisely, we completely solve the Diophantine equation $(x+1)^3 - (x+2)^3 + \cdots - (x + 2d)^3 + (x + 2d + 1)^3 = z^p$, where $p$ is prime and $x,d,z$ are integers with $1 \leq d \leq 50$.
Three integers whose sum, product and the sum of the products of the integers, taken two at a time, are perfect squares
针对欧拉提出的三整数和、积及两两乘积之和均为平方数的问题，本文用新方法得到低次多项式参数化解及数值较小的解。
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 study the equation $F_n + F_m = y^p$, where $F_n$ and $F_m$ are respectively the $n$-th and $m$-th Fibonacci numbers and $p \ge 2$. We find all solutions under the assumption $n \equiv m \pmod{2}$.
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 determine primitive solutions to the equation $(x-r)^2 + x^2 + (x+r)^2 = y^n$ for $1 \le r \le 5,000$, making use of a factorization argument and the Primitive Divisors Theorem due to Bilu, Hanrot and Voutier.
Let us denote by $F_n$ the $n$-th Fibonacci number. In this paper we show that for a fixed integer $y$ there exists at most one integer exponent $a>0$ such that the Diophantine equation $F_n+F_m=y^a$ has a solution $(n,m,a)$ in positive integers satisfying $n>m>0$, unless $y=2,3,4,6$ or $10$.
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…
The sum of the reciprocals of the prime divisors of an odd perfect or odd primitive non-deficient number
本文在奇完全数及更弱的奇本原非亏数假设下，证明了素因子倒数之和 T(n) 与乘积 H(n) 关于最小素因子的新上界。
Let $a\geq 1, b\geq 0$ and $k\geq 2$ be any given integers. It has been proven that there exist infinitely many natural numbers $m$ such that sum of divisors of $m$ is a perfect $k$th power. We try to generalize this result when the values of $m$ belong to any given infinite arithmetic progression $an+b$. We prove if $a$ is relatively prime to $b$ and order of $b$ modulo $a$ is relatively prime to $k$ then there exist infinitely many natural numbers $n$ such that sum of divisors of $an+b$ is a perfect $k$th power. We also prove that, in general, either sum of divisors of $an+b$ is not a perfec…
Much recent progress has been made concerning the probable existence of Odd Perfect Numbers, forming part of what has come to be known as Sylvester's Web Of Conditions. This paper proves some results concerning certain properties of the sums of reciprocals of the factors of odd perfect numbers, or, in more technical terms, the properties of the sub-sums of σ_{-1} (n). By this result, it also establishes strong bounds on the prime factors of odd perfect numbers using the number of distinct prime factors it may possess.
In this paper, we determine all primitive solutions to the equation $(x+r)^2 +(x+2r)^2 +\cdots +(x+dr)^2 = y^n$ for $2\leq d\leq 10$ and for $1\leq r\leq 10^4$. We make use of a factorization argument and the Primitive Divisors Theorem due to Bilu, Hanrot and Voutier.
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.
We extend the sum-of-divisors function to the complex plane via the Gaussian integers. Then we prove a modified form of Euler's classification of odd perfect numbers.
On perfect powers that are sum of two balancing numbers
本文研究平衡数之和构成完全幂的丢番图方程，在同奇偶条件下求出全部正整数解，并进一步考察互素平衡数立方和或差为完全幂的情形。
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…
The study of perfect numbers (numbers which equal the sum of their proper divisors) goes back to antiquity, and is responsible for some of the oldest and most popular conjectures in number theory. We investigate a generalization introduced by Pollack and Shevelev: $k$-near-perfect numbers. These are examples to the well-known pseudoperfect numbers first defined by Sierpiński, and are numbers such that the sum of all but at most $k$ of its proper divisors equals the number. We establish their asymptotic order for all integers $k\ge 4$, as well as some properties of related quantities.
We know that any prime number of form $4s+1$ can be written as a sum of two perfect square numbers. As a consequence of Goldbach's weak conjecture, any number great than $10$ can be represented as a sum of four primes. We are motivated to consider an equation with some constraints about digital sum for the four primes. And we conclude that the square root of the digital sum of the four primes will greater than $4$ and will not be a multiple of $3$ if the equation has solutions. In the proof, we give the method of determining whether a number is a perfect square.
Sum of consecutive powers as a perfect power
本文研究了方程$x^k + (x+1)^k = y^n$（$n\geq 3$）在$k\equiv 2\pmod{4}$时的解，证明当$6\leq k\leq 100$时，唯一解为$x=0$或$x=-1$，或者当$k$的奇素因子满足$k\equiv 3\pmod{4}$时。
We investigate the average number of representations of a positive integer as the sum of $k + 1$ perfect $k$-th powers of primes. We extend recent results of Languasco and the last Author, which dealt with the case $k = 2$ [6] and $k = 3$ [5] respectively. We use the same technique to study the corresponding problem for sums of just $k$ perfect $k$-th powers of primes.
Let $G$ be a finite abelian group and $s$ be a positive integer. A subset $A$ of $G$ is called a {\em perfect $s$-basis of $G$} if each element of $G$ can be written uniquely as the sum of at most $s$ (not-necessarily-distinct) elements of $A$; similarly, we say that $A$ is a {\em perfect restricted $s$-basis of $G$} if each element of $G$ can be written uniquely as the sum of at most $s$ distinct elements of $A$. We prove that perfect $s$-bases exist only in the trivial cases of $s=1$ or $|A|=1$. The situation is different with restricted addition where perfection is more frequent; here we tr…
$q$-Supercongruences from Jackson's $_8ϕ_7$ summation and Watson's $_8ϕ_7$ transformation
本文利用 Jackson $_8ϕ_7$ 求和、Watson $_8ϕ_7$ 变换、创造性显微方法和中国剩余定理，构造分圆多项式五、六次幂模下罕见的 $q$-超同余式，并给出 Long–Ramakrishna 公式的 $q$-模拟及两个双重级数结果。
We generalise the sum-of-divisors-function $σ$ and evenness to the rings of integers of certain algebraic number fields. In particular, we present necessary and sufficient conditions for even Eisenstein integers to be (norm-)perfect based on the work of McDaniel [McD] on Gaussian integers. Furthermore, some results concerning odd norm-perfect Eisenstein integers and the rings of integers of other cyclotomic fields are proven.
Let $σ(n)$ denote the sum of the positive divisors of $n$. We say that $n$ is perfect if $σ(n) = 2 n$. Currently there are no known odd perfect numbers. It is known that if an odd perfect number exists, then it must be of the form $N = p^α\prod_{j=1}^k q_j^{2 β_j}$, where $p, q_1, ..., q_k$ are distinct primes and $p \equiv α\equiv 1 \pmod{4}$. Define the total number of prime factors of $N$ as $Ω(N) := α+ 2 \sum_{j=1}^k β_j$. Sayers showed that $Ω(N) \geq 29$. This was later extended by Iannucci and Sorli to show that $Ω(N) \geq 37$. This was extended by the author to show that $Ω(N) \geq 47$…
In this paper, we introduce the concept of $F$-perfect number, which is a positive integer $n$ such that $\sum_{d|n,d<n}d^2=3n$. We prove that all the $F$-perfect numbers are of the form $n=F_{2k-1}F_{2k+1}$, where both $F_{2k-1}$ and $F_{2k+1}$ are Fibonacci primes. Moreover, we obtain other interesting results and raise a new conjecture on perfect numbers.
A positive integer n is said to be perfect if sigma(n)=2n, where sigma denotes the sum of the divisors of n. In this article, we show that if n is an even perfect number, then any integer m<=n is expressed as a sum of some of divisors of n.
On the Asymptotic Behavior of a Multiplicative Arithmetic Function Related to the Divisor Function Over Perfect Squares Integers Generated by Shifting
本文研究了与完美平方整数相关的除数函数的渐进行为，推导了特定求和式的渐近公式。
A number is perfect if it is the sum of its proper divisors; here we call a finite group `perfect' if its order is the sum of the orders of its proper normal subgroups. (This conflicts with standard terminology but confusion should not arise.) The notion of perfect group generalizes that of perfect number, since a cyclic group is perfect exactly when its order is perfect. We show that, in fact, the only abelian perfect groups are the cyclic ones, and exhibit some non-abelian perfect groups of even order.
Perfect powers in polynomial power sums
本文证明了阶数至少为二的非退化简单线性递推多项式序列不能包含任意大的多项式幂，且在二阶情况下存在索引上界限制元素成为真正的幂。
We develop a multigrade enhancement of the efficient congruencing method to estimate Vinogradov's integral of degree $k$ for moments of order $2s$, thereby obtaining near-optimal estimates for $\tfrac{5}{8}k^2<s\le k^2-k+1$. There are numerous applications. In particular, when $k$ is large, the anticipated asymptotic formula in Waring's problem is established for sums of $s$ $k$th powers of natural numbers whenever $s>1.543k^2$. The asymptotic formula is also established for sums of $28$ fifth powers.
A perfect number is a positive integer n such that n equals the sum of all positive integer divisors of n that are less than n. That is, although n is a divisor of n, n is excluded from this sum. Thus 6 = 1 + 2 + 3 is perfect, but 12 < 1 + 2 + 3 + 4 + 6 is not perfect. An ACL2 theory of perfect numbers is developed and used to prove, in ACL2(r), this bit of mathematical folklore: Even if there are infinitely many perfect numbers the series of the reciprocals of all perfect numbers converges.
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。
In a recent paper we proved that if (*)=\inf_{|z_k|=1}\max_{v=1,...,n^2-n} |\sum_{k=1}^n z_k^v|, then (*)=\sqrt{n-1} if n-1 is a prime power. We proved that a construction of Fabrykowski gives minimal systems (z_1,...,z_n) to this problem. The construction depends on the existence of perfect difference sets of order n-1. As an open problem we asked whether all solutions would arise from this construction. In this paper we show that this is true and in fact if there exist no perfect difference set of order n-1 (which by the prime power conjecture is true if n-1 is not a prime power), then we ha…
Product representations of perfect powers
本文研究了完美幂的乘积表示问题，证明了在特定条件下，最大集合的大小与素数幂相关函数的和有关，解决了Verstraëte提出的问题。
On pairs of triangular numbers whose product is a perfect square and pairs of intervals of successive integers with equal sums of squares
本文研究了乘积为完全平方数的三角数对以及具有相等平方和的连续整数区间对，构造了多项式并提出了相关猜想。
Let $p\ge 3$ be a prime, $S\subseteq\mathbb F_p^2$ a nonempty set, and $w\colon\mathbb F_p^2\to\mathbb R$ a function with $\mathrm{supp}\, w=S$. Applying an uncertainty inequality due to András Biró and the present author, we show that there are at most $\frac12|S|$ directions in $\mathbb F_p^2$ such that for every line $l$ in any of these directions, one has $$ \sum_{z\in l} w(z) = \frac1p\sum_{z\in\mathbb F_p^2} w(z), $$ except if $S$ itself is a line and $w$ is constant on $S$ (in which case all, but one direction have the property in question). The bound $\frac12|S|$ is sharp. As an applic…
Compact formulas are obtained for the Casimir energy of a relativistic perfect fluid confined to a $D$-dimensional hypercube with von Neumann or Dirichlet boundary conditions. The formulas are conveniently expressed as a finite sum of the well-known gamma and Riemann zeta functions. Emphasis is placed on the mathematical technique used to extract the Casimir energy from a $D$-dimensional infinite sum regularized with an exponential cut-off. Numerical calculations show that initially the Dirichlet energy decreases rapidly in magnitude and oscillates in sign, being positive for even $D$ and nega…
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