Related papers: The Sum of Squares function in the ring $\mathbb{Z…
We know that $\mathbb{Z}_n$ is a finite field for a prime number $n$. Let $m,n$ be arbitrary natural numbers and let $\mathbb{Z}^m_n= \mathbb{Z}_n \times\mathbb{Z}_n\times...\times\mathbb{Z}_n$ be the Cartesian product of $m$ rings…
We prove explicit bounds for the number of sums of consecutive prime squares below a given magnitude.
Counting functions are constructed for sums of integers raised to a fixed positive rational power. That is, given values formed by $u_1^{j/k} + u_2^{j/k} + ... + u_l^{j/k}$, $u_i \in \mathbb{Z}^+$, the number of values less than or equal to…
Finite trigonometric sums appear in various branches of Physics, Mathematics and their applications. For p; q to coprime positive integers and r we consider the finite trigonometric sums involving the product of three trigonometric…
Let $P(x) \in \mathbb{Z}[x]$ be a polynomial. We give an easy and new proof of the fact that the set of primes $p$ such that $p \mid P(n)$, for some $n \in \mathbb{Z}$, is infinite. We also get analog of this result for some special…
Let $m,n\in \Bbb{N}$. We represent the additive subgroups of the ring $\Bbb{Z}_m \times \Bbb{Z}_n$, which are also (unital) subrings, and deduce explicit formulas for $N^{(s)}(m,n)$ and $N^{(us)}(m,n)$, denoting the number of subrings of…
We examine oscillations in a number of sums of arithmetic functions involving $\Omega(n)$, the total number of prime factors of $n$, and $\omega(n)$, the number of distinct prime factors of $n$. In particular, we examine oscillations in…
We develop a theory of multiplicative functions (with values inside or on the unit circle) in arithmetic progressions analogous to the well-known theory of primes in arithmetic progressions.
We determine all perfect powers that can be written as the sum of at most 10 consecutive squares.
This is the first in a series of three papers dealing with sums of squares and hypoellipticity in the infinite regime. We give a sharp sufficient condition on a smooth nonnegative function f on n-dimensional Euclidean space so that it can…
Some class of sums which naturally include the sums of powers of integers is considered. A number of conjectures concerning a representation of these sums is made.
We study the ring of polyfunctions over $\mathbb Z/n\mathbb Z$. The ring of polyfunctions over a commutative ring $R$ with unit element is the ring of functions $f:R\to R$ which admit a polynomial representative $p\in R[x]$ in the sense…
The problem of finding the sum of a polynomial's values is considered. In particular, for any $n\geq 3$, the explicit formula for the sum of the $n$th powers of natural numbers $S_n=\sum_{x=1}^{m}x^{n}$ is proved:…
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…
We define a fat staircase to be a Ferrers diagram corresponding to a partition of the form $(n^{\alpha_n}, {n-1}^{\alpha_{n-1}},..., 1^{\alpha_1})$, where $\alpha = (\alpha_1,...,\alpha_n)$ is a composition, or the $180^\circ$ rotation of…
A unitary divisor $c$ of a positive integer $n$ is a positive divisor of $n$ that is relatively prime to $\displaystyle{\frac{n}{c}}$. For any integer $k$, the function $\sigma_k^*$ is a multiplicative arithmetic function defined so that…
We show that when projecting an edge-transitive N-dimensional polytope onto anM-dimensional subspace of R^N, the sums of the squares of the original and projected edges are in the ratio N=M.
This paper gives new explicit formulas for sums of powers of integers and their reciprocals.
Using Fermat's two squares theorem and properties of cyclotomic polynomials, we prove assertions about when numbers of the form $a^{n}+1$ can be expressed as the sum of two integer squares. We prove that $a^n + 1$ is the sum of two squares…
Recent results about sums of cubes of Fibonacci numbers [Frontczak, 2018] are extended to arbitrary powers.