Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userDiophantine equations with sum of cubes and cube of sum
269
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
We solve Diophantine equations of the type $ , a , (x^3 + y^3 + z^3 ) = (x + y + z)^3$, where $x,y,z$ are integer variables, and the coefficient $a eq 0$ is rational. We show that there are infinite families of such equations, including those where $a$ is any ratio of cubes or certain rational fractions, that have nontrivial solutions. There are also infinite families of equations that do not have any nontrivial solution, including those where $1/a = 1 - 24/m$ with certain restrictions on the integer $m$. The equations can be represented by elliptic curves unless $a = 9$ or 1. If $a$ is an integer and two variables are equal and nonzero, there exist nontrivial solutions only for $a=4$ or 9; there are no solutions for $a = 4$ when $xyz eq 0$. Without imposing constraints on the variables, we find the general solution for $a = 9$, which depends on two integer parameters. These cubic equations are important in particle physics, because they determine the fermion charges under the $U(1)$ gauge group.
rate research
Read More
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 $1 leq d leq 50$.
Divisor functions have attracted the attention of number theorists from Dirichlet to the present day. Here we consider associated divisor functions $c_j^{(r)}(n)$ which for non-negative integers $j, r$ count the number of ways of representing $n$ as an ordered product of $j+r$ factors, of which the first $j$ must be non-trivial, and their natural extension to negative integers $r.$ We give recurrence properties and explicit formulae for these novel arithmetic functions. Specifically, the functions $c_j^{(-j)}(n)$ count, up to a sign, the number of ordered factorisations of $n$ into $j$ square-free non-trivial factors. These functions are related to a modified version of the Mobius function and turn out to play a central role in counting the number of sum systems of given dimensions. par Sum systems are finite collections of finite sets of non-negative integers, of prescribed cardinalities, such that their set sum generates consecutive integers without repetitions. Using a recently established bijection between sum systems and joint ordered factorisations of their component set cardinalities, we prove a formula expressing the number of different sum systems in terms of associated divisor functions.
Let $gcd(k,j)$ denote the greatest common divisor of the integers $k$ and $j$, and let $r$ be any fixed positive integer. Define $$ M_r(x; f) := sum_{kleq x}frac{1}{k^{r+1}}sum_{j=1}^{k}j^{r}f(gcd(j,k)) $$ for any large real number $xgeq 5$, where $f$ is any arithmetical function. Let $phi$, and $psi$ denote the Euler totient and the Dedekind function, respectively. In this paper, we refine asymptotic expansions of $M_r(x; {rm id})$, $M_r(x;{phi})$ and $M_r(x;{psi})$. Furthermore, under the Riemann Hypothesis and the simplicity of zeros of the Riemann zeta-function, we establish the asymptotic formula of $M_r(x;{rm id})$ for any large positive number $x>5$ satisfying $x=[x]+frac{1}{2}$.
We investigate various questions concerning the reciprocal sum of divisors, or prime divisors, of the Mersenne numbers $2^n-1$. Conditional on the Elliott-Halberstam Conjecture and the Generalized Riemann Hypothesis, we determine $max_{nle x} sum_{p mid 2^n-1} 1/p$ to within $o(1)$ and $max_{nle x} sum_{dmid 2^n-1}1/d$ to within a factor of $1+o(1)$, as $xtoinfty$. This refines, conditionally, earlier estimates of ErdH{o}s and ErdH{o}s-Kiss-Pomerance. Conditionally (only) on GRH, we also determine $sum 1/d$ to within a factor of $1+o(1)$ where $d$ runs over all numbers dividing $2^n-1$ for some $nle x$. This conditionally confirms a conjecture of Pomerance and answers a question of Murty-Rosen-Silverman. Finally, we show that both $sum_{pmid 2^n-1} 1/p$ and $sum_{dmid 2^n-1}1/d$ admit continuous distribution functions in the sense of probabilistic number theory.
For $ Esubset mathbb{F}_q^d$, let $Delta(E)$ denote the distance set determined by pairs of points in $E$. By using additive energies of sets on a paraboloid, Koh, Pham, Shen, and Vinh (2020) proved that if $E,Fsubset mathbb{F}_q^d $ are subsets with $|E||F|gg q^{d+frac{1}{3}}$ then $|Delta(E)+Delta(F)|> q/2$. They also proved that the threshold $q^{d+frac{1}{3}}$ is sharp when $|E|=|F|$. In this paper, we provide an improvement of this result in the unbalanced case, which is essentially sharp in odd dimensions. The most important tool in our proofs is an optimal $L^2$ restriction theorem for the sphere of zero radius.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
