Related papers: Seeds for Generalized Taxicab Numbers

Consider the linear congruence equation $${a_1^{s}x_1+\ldots+a_k^{s} x_k \equiv b\,(\text{mod } n^s)}\text { where } a_i,b\in\mathbb{Z},s\in\mathbb{N}$$ Denote by $(a,b)_s$ the largest $l^s\in\mathbb{N}$ which divides $a$ and $b$…

A Universal Cycle for t-multisets of [n]={1,...,n} is a cyclic sequence of $\binom{n+t-1}{t}$ integers from [n] with the property that each t-multiset of [n] appears exactly once consecutively in the sequence. For such a sequence to exist…

In this paper we confirm a conjecture of Sun which states that each positive integer is a sum of a square, an odd square and a triangular number. Given any positive integer m, we show that p=2m+1 is a prime congruent to 3 modulo 4 if and…

The Stern sequence (s(n)) is defined by s(0) = 0, s(1) = 1, s(2n) = s(n), s(2n+1) = s(n) + s(n+1). Stern showed in 1858 that gcd(s(n),s(n+1)) = 1, and that for every pair of relatively prime positive integers (a,b), there exists a unique n…

For each positive integer n greater than or equal to 2, a new approach to expressing real numbers as sequences of nonnegative integers is given. The n=2 case is equivalent to the standard continued fraction algorithm. For n=3, it reduces to…

For an integer \( k \geq 2 \), the sequence of \( k \)-generalized Lucas numbers is defined by the recurrence relation \( L_n^{(k)} = L_{n-1}^{(k)} + \cdots + L_{n-k}^{(k)} \) for all \( n \geq 2 \), with initial conditions \( L_0^{(k)} = 2…

For two graph H and G, the Ramsey number r(H, G) is the smallest positive integer n such that every red-blue edge coloring of the complete graph K_n on n vertices contains either a red copy of H or a blue copy of G. Motivated by questions…

We study the generalized Ramsey numbers $f(Q_n, C_{k}, q)$, that is, the minimum number of colors needed to edge-color the hypercube $Q_n$ so that every copy of the cycle $C_{k}$ has at least $q$ colors. Our main result is that for any…

For given graphs $G$ and $H,$ the \emph{Ramsey number} $R(G,H)$ is the least natural number $n$ such that for every graph $F$ of order $n$ the following condition holds: either $F$ contains $G$ or the complement of $F$ contains $H.$ In this…

For coprime positive integers $q$ and $e$, let $m(q,e)$ denote the least positive integer $t$ such that there exists a sum of $t$ powers of $q$ which is divisible by $e$. We prove an upper bound for $m(q.e)$ and investigate the case where…

Let $a$ and $b$ be relatively prime integers. Then the first Lucas sequence $\left(U_n\right)_{n\geq0}$ and the second Lucas sequence $\left(V_n\right)_{n\geq0}$ are defined respectively by $U_{n+2}=aU_{n+1}+bU_{n},\, U_0=0,\,U_1=1$ and…

Let $G(k)$ denote the least number $s$ having the property that every sufficiently large natural number is the sum of at most $s$ positive integral $k$-th powers. Then for all $k\in \mathbb N$, one has \[ G(k)\le \lceil k(\log…

For a subset $S$ of nonnegative integers and a vector $\mathbf{a}=(a_1,\dots,a_k)$ of positive integers, let $V'_S(\mathbf{a})=\{ a_1s_1+\cdots+a_ks_k : s_i\in S\} \setminus \{0\}$. For a positive integer $n$, let $\mathcal T(n)$ be the set…

Given graphs $G, H$ and an integer $q \ge 2$, the generalized Ramsey number, denoted $r(G,H,q)$, is the minimum number of colours needed to edge-colour $G$ such that every copy of $H$ receives at least $q$ colours. In this paper, we prove…

The Alon-Tarsi number $AT(G)$ of a graph $G$ is the smallest $k$ for which there is an orientation $D$ of $G$ with max indegree $k-1$ such that the number of even and odd circulations contained in D are different. In this paper, we show…

An integer of the form $P_m(x)= \frac{(m-2)x^2-(m-4)x}{2}$ for an integer $x$, is called a generalized $m$-gonal number. For positive integers $\alpha_1,\dots,\alpha_u$ and $\beta_1,\dots,\beta_v$, a mixed sum…

The cage problem asks for the smallest number $c(k,g)$ of vertices in a $k$-regular graph of girth $g$ and graphs meeting this bound are known as cages. While cages are known to exist for all integers $k \ge 2$ and $g \ge 3$, the exact…

For positive integers $a,b,c$, and an integer $n$, the number of integer solutions $(x,y,z) \in \mathbb Z^3$ of $a \frac{x(x-1)}{2} + b \frac{y(y-1)}{2} + c \frac{z(z-1)}{2} = n$ is denoted by $t(a,b,c;n)$. In this article, we prove some…

Let $S\subseteq V(G)$ and $\kappa_{G}(S)$ denote the maximum number $r$ of edge-disjoint trees $T_1, T_2, \cdots, T_r$ in $G$ such that $V(T_i)\bigcap V(T_{j})=S$ for any $i, j \in \{1, 2, \cdots, r\}$ and $i\neq j$. For an integer $k$ with…

The well know theorem of Tverberg states that if n > (d+1)(r-1) then one can partition any set of n points in R^d to r disjoint subsets whose convex hulls have a common point. The numbers T(d,r) = (d + 1)(r - 1) + 1 are known as Tverberg…