Related papers: Sum of consecutive powers as a perfect power

We find all positive integer solutions in $x, y$ and $n$ of $x^2+19^{2k+1}=4y^{n}$ for any non-negative integer $k$.

Let $p^k m^2$ be an odd perfect number with special prime $p$. In this article, we provide an alternative proof for the biconditional that $\sigma(m^2) \equiv 1 \pmod 4$ holds if and only if $p \equiv k \pmod 8$. We then give an application…

We prove a conjecture posted in the Online Encyclopedia of Integer Sequences, namely that there are exactly five positive integers that can be written in more than one way as the sum of a nonnegative power of 2 and a nonnegative power of 3.…

Let $w_{n+2}=pw_{n+1}+qw_{n}$ for $n\geq0$ with $w_0=a$ and $w_1=b$. In this paper we find an explicit expression, in terms of determinants, for $\sum_{n\geq0} w_n^kx^n$ for any $k\geq1$. As a consequence, we derive all the previously known…

We determine all perfect powers that can be written as the sum of at most 10 consecutive squares.

Let $P,Q\in\Bbb Z$, $U_0=0,\ U_1=1$ and $U_{n+1}=PU_n-QU_{n+1}$. In this paper we obtain a general congruence for $U_{kmn^r}/U_k\pmod {n^{r+1}}$, where $k,m,n,r$ are positive integers. As applications we extend Lucas' law of repetition and…

The Delannoy polynomial $D_n(x)$ is defined by $$ D_n(x)=\sum_{k=0}^{n}{n\choose k}{n+k\choose k}x^k. $$ We prove that, if $x$ is an integer and $p$ is a prime not dividing $x(x+1)$, then \begin{align*} \sum_{k=0}^{p-1}(2k+1)D_k(x)^3…

Let $p$ be an odd prime, Jianqiang Zhao has established a curious congruence, which is $$ \sum_{i+j+k=p \atop i,j,k > 0} \frac{1}{ijk} \equiv -2B_{p-3}\pmod p , $$ where $B_{n}$ denotes the $n$-th Bernoulli number. In this paper, we will…

The power sum $1^n + 2^n + \cdots + x^n$ has been of interest to mathematicians since classical times. Johann Faulhaber, Jacob Bernoulli, and others who followed expressed power sums as polynomials in $x$ of degree $n+1$ with rational…

We study the Goldbach problem for primes represented by the polynomial $x^2+y^2+1$. The set of such primes is sparse in the set of all primes, but the infinitude of such primes was established by Linnik. We prove that almost all even…

A natural number is a binary $k$'th power if its binary representation consists of $k$ consecutive identical blocks. We prove an analogue of Waring's theorem for sums of binary $k$'th powers. More precisely, we show that for each integer $k…

We study two new classes of sums with inverse binomial coefficients and harmonic numbers. In addition we establish recursive solutions to the following power sums \begin{equation*} U_d(n) = \sum_{k=1}^n \frac{2^{2k}}{\binom{2k}{k}} \cdot…

Let $\Lambda(n)$ be the von Mangoldt function, $x$ real and $2\leq y \leq x$. This paper improves the estimate on the exponential sum over primes in short intervals \[ S_k(x,y;\alpha) = \sum_{x< n \leq x+y} \Lambda(n) e\left( n^k \alpha…

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…

An open conjecture of Erdos and Moser is that the only solution of the Diophantine equation in the title is the trivial solution 1+2=3. Reducing the equation modulo k and k^2, we give necessary and sufficient conditions on solutions to the…

For $n \geq 3$, an asymptotic formula is derived for the number of representations of a sufficiently large natural number $N$ as a sum of $r = 2^n + 1$ summands, each of which is an $n$-th power of natural numbers $x_i$, $i = \overline{1,…

We obtain solutions to the recursive sequences of the form $$x_{n + 1} = \frac{x_{n - 3}x_{n }}{x_{n - 2}(a_n + b_nx_{n -3}x_{n})}$$ where $a_n$ and $b_n$ are arbitrary sequences of real numbers, and the initial values are gives as;…

This paper contributes to the conjecture of R. Scott and R. Styer which asserts that for any fixed relatively prime positive integers $a,b$ and $c$ all greater than 1 there is at most one solution to the equation $a^x+b^y=c^z$ in positive…

Let $p_{k,3}(n)$ enumerate the number of 2-color partition triples of $n$ where one of the colors appears only in parts that are multiples of $k$. In this paper, we prove several infinite families of congruences modulo powers of 3 for…

We address three questions posed by Bibak \cite{KB20}, and generalize some results of Bibak, Lehmer and K G Ramanathan on solutions of linear congruences $\sum_{i=1}^k a_i x_i \equiv b \Mod{n}$. In particular, we obtain explicit expressions…