Related papers: Sums of even ascending powers

Drawing inspiration from the work of Nathanson and Yamada we prove that every even integer larger than $\exp (\exp (32.7))$ can be written as the sum of a prime and the product of at most two primes.

We give an expression of polynomials for higher sums of powers of integers via the higher order Bernoulli numbers.

It is proved that all sufficiently large integers $n$ can be represented as $$n=x_1^2+x_2^3+\cdots+x_{13}^{14},$$ where $x_1,\ldots,x_{13}$ are positive integers. This improves upon the current record with $14$ variables in place of $13$.

We present and analyze an algorithm to enumerate all integers $n\le x$ that can be written as the sum of consecutive $k$th powers of primes, for $k>1$. We show that the number of such integers $n$ is asymptotically bounded by a constant…

Recently, E. Samsonadze (arXiv:2411.11859v1) has given an explicit formula for the sums of powers of integers $S_k(n) = 1^k +2^k +\cdots + n^k$. In this short note, we show that Samsonadze's formula corresponds to a well-known formula for…

Erdos conjectured that every odd number greater than one can be expressed as the sum of a squarefree number and a power of two. Subsequently, Odlyzko and McCranie provided numerical verification of this conjecture up to $10^7$ and $1.4\cdot…

In 1951, Linnik proved the existence of a constant $K$ such that every sufficiently large even number is the sum of two primes and at most $K$ powers of 2. Since then, this style of approximation has been considered for problems similar to…

In this paper, we derive a formula for the sums of powers of the first $n$ positive integers, $S_k(n)$, that involves the hyperharmonic numbers and the Stirling numbers of the second kind. Then, using an explicit representation for the…

Let \sigma(n) be the sum of divisors of a positive integer n. Robin's theorem states that the Riemann hypothesis is equivalent to the inequality \sigma(n)<e^\gamma n\log\log n for all n>5040 (\gamma is Euler's constant). It is a natural…

In this paper, we consider representations of integers as sums of generalized heptagonal numbers with a prescribed number of repeats of each heptagonal number appearing in the sum. In particular, we investigate the classification of such…

A classical theorem of Paley asserts the existence of an infinite family of quadratic characters whose character sums become exceptionally large. In this paper, we establish an analogous result for characters of any fixed even order.…

This paper gives new explicit formulas for sums of powers of integers and their reciprocals.

An positive even number is said to be a Kronecker number if it can be written in infinitely many ways as the difference between two primes, and it is believed that all even numbers are Kronecker numbers. We will study the division and…

We prove that if A is a subset of the primes, and the lower density of A in the primes is larger than 1/2, then every sufficiently large even integer can be written as the sum of eight primes from A. The constant 1/2 in this statement is…

We approach a new proof of the strong Goldbach's conjecture for sufficiently large even integers by applying the Dirichlet's series. Using the Perron formula and the Residue Theorem in complex variable integration, one could show that any…

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…

We proved that any even number not less than 6 can be expressed as the sum of two old primes, $2n=p_i+p_j$

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 $k = 3$…

We give an upper bound for the exponential sum over squarefree integers. This establishes a conjecture by Br\"udern and Perelli.

By extending a construction due to Gross and McMullen [2], we show that for any odd integer n and for any even integer d>n+2 there are infinitely many Salem numbers $\alpha$ of degree d such that $\alpha^n-1$ is a unit. A similar result is…