Related papers: On gaps between sums of four fourth powers

Let ${\mathcal H}$ be a multiplicative subgroup of $\mathbb{F}_p^*$ of order $H>p^{1/4}$. We show that $$ \max_{(a,p)=1}\left|\sum_{x\in {\mathcal H}} {\mathbf{\,e}}_p(ax)\right| \le H^{1-31/2880+o(1)}, $$ where ${\mathbf{\,e}}_p(z) =…

The $4$-girth-thickness $\theta(4,G)$ of a graph $G$ is the minimum number of planar subgraphs of girth at least four whose union is $G$. In this paper, we obtain that the 4-girth-thickness of complete tripartite graph $K_{n,n,n}$ is…

We construct the effective action and gap equations for nonperturbative fermion 4-point functions. Our results apply to situations in which fermion masses can be ignored, which is the case for theories of strong flavor interactions…

We establish an upper bound for the rank of every power of an arbitrary quadratic form. Specifically, for any $s\in\mathbb{N}$, we prove that the $s$-th power of a quadratic form of rank $n$ grows as $n^s$. Furthermore, we demonstrate that…

In this paper we study mixed sums of primes and linear recurrences. We show that if m=2(mod 4) and m+1 is a prime then $(m^{2^n-1}-1)/(m-1)\not=m^n+p^a$ for any n=3,4,... and prime power p^a. We also prove that if a>1 is an integer, u_0=0,…

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.

By applying the Newton-Gregory expansion to the polynomial associated with the sum of powers of integers $S_k(n) = 1^k + 2^k + \cdots + n^k$, we derive a couple of infinite families of explicit formulas for $S_k(n)$. One of the families…

Using Watt's mean value theorem and a delicate sieve decomposition, the author shows that the interval $[x, x+x^{\frac{1}{2}+\varepsilon}]$ contains an integer with a prime factor larger than $x^{\frac{35}{36}-\varepsilon}$ for sufficiently…

Translated from the Latin original, "Observationes circa bina biquadrata quorum summam in duo alia biquadrata resolvere liceat" (1772). E428 in the Enestroem index. This paper is about finding A,B,C,D such that $A^4+B^4=C^4+D^4$. In sect.…

An asymptotic expansion for the generalised quadratic Gauss sum $$S_N(x,\theta)=\sum_{j=1}^{N} \exp (\pi ixj^2+2\pi ij\theta),$$ where $x$, $\theta$ are real and $N$ is a positive integer, is obtained as $x\rightarrow 0$ and…

Let $f:\Z/q\Z\rightarrow\Z$ be such that $f(a)=\pm 1$ for $1\le a<q$, and $f(q)=0$. Then Erd\"os conjectured that $\sum_{n\ge1}\frac{f(n)}{n} \ne 0$. For $q$ even, this is trivially true. If $q\equiv 3$ ( mod $4$), Murty and Saradha proved…

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

Let $P$ be a subset of the primes of lower density strictly larger than $\frac12$. Then, every sufficiently large even integer is a sum of four primes from the set $P$. We establish similar results for $k$-summands, with $k\geq 4$, and for…

An N=4 supersymmetric extension of the l-conformal Galilei algebra is constructed. This is achieved by combining generators of spatial symmetries from the l-conformal Galilei algebra and those underlying the most general superconformal…

A famous conjecture of Erd\H os and Straus is that for every integer $n\ge2$, $4/n$ can be represented as $1/x+1/y+1/z$, where $x,y,z$ are positive integers. This conjecture was generalized to $5/n$ by Sierpi\'nski, and then Schinzel…

We extend a formula of Duke, Imam\=oglu, and T\'oth (which itself is a generalization of the Katok-Sarnak formula) to prove the Kohnen-Zagier formula for Maass forms for $\Gamma_0(4)$.

For integer $m, p,$ we study tangent power sum $\sum^m_{k=1}\tan^{2p}\frac{\pi k}{2m+1}.$ We prove that, for every $m, p,$ it is integer, and, for a fixed p, it is a polynomial in $m$ of degree $2p.$ We give recurrent, asymptotical and…

Let $G$ be a complete $k$-partite simple undirected graph with parts of sizes $p_1\le p_2...\le p_k$. Let $P_j=\sum_{i=1}^jp_i$ for $j=1,...,k$. It is conjectured that $G$ has distance magic labeling if and only if $\sum_{i=1}^{P_j}…

We establish some supercongruences related to a supercongruence of Van Hamme, such as \begin{align*} \sum_{k=0}^{(p+1)/2} (-1)^k (4k-1)\frac{(-\frac{1}{2})_k^3}{k!^3} &\equiv p(-1)^{(p+1)/2}+p^3(2-E_{p-3})\pmod{p^{4}},\\…

We characterise the one-loop amplitudes for N=6 and N=4 supergravity in four dimensions. For N=6 we find that the one-loop n-point amplitudes can be expanded in terms of scalar box and triangle functions only. This simplification is…