Related papers: On sums of three squares

In this paper, we use a modular form approach to evaluate the convolution sums $\sum_{l+42m=n}\sigma (l)\sigma (m)$, $\sum_{2l+21m=n}\sigma (l)\sigma (m),$ $\sum_{3l+14m=n}\sigma (l)\sigma (m)$ and $\sum_{6l+7m=n}\sigma (l)\sigma (m) $ for…

For any sufficiently large $\ell$, suppose that $\ell$ can be expressed as $$ \ell=p_1^4+p_2^4+p_3^4+ \cdots +p_8^4,$$ where $p_1, p_2,p_3,\cdots, p_8$ are primes.For such $\ell$, in this paper we will use circle method and sieves to prove…

We use sums of Liouville type to count the number of ways a positive integer can be represented by the forms $(a+c)^{1/3}x + (b+d)y$, $(a+c)x + \bigl(k(b+d) \bigr)^{1/3} y$, and $\bigl(k(a+c) \bigr)^{1/3} x + l(b+d) y$ for nonnegative…

For $n,\,d\ge1$ let $p(n,2d)$ denote the smallest number $p$ such that every sum of squares of forms of degree $d$ in $\mathbb{R}[x_1,\dots,x_n]$ is a sum of $p$ squares. We establish lower bounds for these numbers that are considerably…

We evaluate the convolution sums $\sum_{l,m\in {\mathbb N}, {l+15m=n}} \sigma(l) \sigma(m)$ and $\sum_{l,m\in {\mathbb N}, {3l+5m=n}} \sigma(l) \sigma(m)$ for all $n\in {\mathbb N}$ using the theory of quasimodular forms and use these…

We evaluate the convolution sum $\displaystyle W_{a,b}(n):= \sum_{al+bm=n} \hspace{-3mm} \sigma(l) \sigma(m)$ for $(a,b)=(1,28), (4,7), (2,7)$ for all positive integers $n$. We use a modular form approach. We also re-evaluate the known sums…

Let $m$ be a positive integer and $b_{m}(n)$ be the number of partitions of $n$ with parts being powers of 2, where each part can take $m$ colors. We show that if $m=2^{k}-1$, then there exists the natural density of integers $n$ such that…

We establish that, for almost all natural numbers $N$, there is a sum of two positive integral cubes lying in the interval $[N-N^{7/18+\epsilon},N]$. Here, the exponent $7/18$ lies half way between the trivial exponent $4/9$ stemming from…

We establish a lower bound for the frequency with which an irreducible monic cubic polynomial with negative discriminant can be expressed as a sum of two squares ($\square_{2}$). This provides a quantitative answer to a question posed by…

Let $a_k(n)$ denotes the number of representations of a non-negative integer $n$ as sum of $k$ quadratic forms of the type $x^2+xy+y^2$ and $a_{\lambda_1,\lambda_2,\lambda_3\dots\lambda_k}(n)$ denotes the number of representations $n$ as a…

We survey the potential for progress in additive number theory arising from recent advances concerning major arc bounds associated with mean value estimates for smooth Weyl sums. We focus attention on the problem of representing large…

We prove that every integer $n \geq 10$ such that $n \not\equiv 1 \text{mod} 4$ can be written as the sum of the square of a prime and a square-free number. This makes explicit a theorem of Erd\H{o}s that every sufficiently large integer of…

In this paper, it is proved that every sufficiently large even integer can be represented as the sum of two squares of primes, two cubes of primes, two biquadrates of primes and 16 powers of 2. Furthermore, there are at least 5.313% odd…

In this paper, we evaluate a smoothed sum of the form $\displaystyle \sideset{}{^*}\sum_{d \leq X}\left(\sum_{n \leq Y} \left(\frac{8d}{n} \right ) \right)^2$, where $\left ( \frac{8d}{\cdot} \right )$ is the Kronecker symbol and…

We obtain new Poisson type summation formulas with nodes $\pm \sqrt{n}$ and with weights involving the function $r_k(n)$ that gives the number of representations of a positive integer $n$ as the sum of $k$ squares. Our results extend…

Translated from the Latin original "Facillima methodus plurimos numeros primos praemagnos inveniendi" (1778). E718 in the Enestrom index. If m is a number of the form 4k+1 and is a sum of two relatively prime squares, then it is prime if…

In this paper we give unified formulas for the numbers of representations of positive integers as sums of four generalized $m$-gonal numbers, and as restricted sums of four squares under a linear condition, respectively. These formulas are…

Let $\{E_n\}$ be the Euler numbers. In the paper we determine $E_{2^mk+b}-E_b$ modulo $2^{m+7}$, where $k$ and $m$ are positive integers and $b\in{0,2,4,...}$.

Orthogonal designs and weighing matrices have many applications in areas such as coding theory, cryptography, wireless networking and communication. In this paper, we first show that if positive integer $k$ cannot be written as the sum of…

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…