kth Power of a Partial Sum

ON kTH-POWER NUMERICAL CENTRES

We call the integer N a kth-power numerical centre for n if 1 + 2 + · · · + N = N + (N + 1) + · · · + n. We prove, using the explicit lower bounds on linear forms in elliptic logarithms, that there are no nontrivial fifth-power numerical centres for any n, and demonstrate that there are only finitely many pairs (N, n) satisfying the above for any given k > 1. The problem of finding kth-power ce...

kTH POWER RESIDUE CHAINS OF GLOBAL FIELDS

In 1974, Vegh proved that if k is a prime and m a positive integer, there is an m term permutation chain of kth power residue for infinitely many primes [E.Vegh, kth power residue chains, J.Number Theory, 9(1977), 179-181]. In fact, his proof showed that 1, 2, 22, ..., 2m−1 is an m term permutation chain of kth power residue for infinitely many primes. In this paper, we prove that for k being a...

The Wiener index of the kth power of a graph

The kth power of a graph G, denoted by Gk , is a graph with the same vertex set as G such that two vertices are adjacent in Gk if and only if their distance is at most k in G. The Wiener index is a distance-based topological index defined as the sum of distances between all pairs of vertices in a graph. In this note, we give the bounds on the Wiener index of the graph Gk . The Nordhaus–Gaddum-t...

Perfectness and Imperfectness of the kth Power of Lattice Graphs

Given a pair of non-negative integers m and n, S(m,n) denotes a square lattice graph with a vertex set {0, 1, 2, . . . ,m − 1} × {0, 1, 2, . . . , n− 1}, where a pair of two vertices is adjacent if and only if the distance is equal to 1. A triangular lattice graph T (m,n) has a vertex set {(xe1 + ye2) | x ∈ {0, 1, 2, . . . ,m− 1}, y ∈ {0, 1, 2, . . . , n− 1}} where e1 def. = (1, 0), e2 def. = (...

The Wiener Polynomial of the kth Power Graph