The k-tuple jumping champions among consecutive primes

Jumping Champions and Gaps between Consecutive Primes

For any real x, the most common difference that occurs among the consecutive primes less than or equal to x is called a jumping champion. This term was introduced by J. H. Conway in 1993. There are occasionally ties. Therefore there can be more than one jumping champion for a given x. The first, but short-lived, jumping champion is 1. Aside from the numerical studies, nothing else has been prov...

Jumping Champions

k-TUPLE DOMATIC IN GRAPHS

For every positive integer k, a set S of vertices in a graph G = (V;E) is a k- tuple dominating set of G if every vertex of V -S is adjacent to at least k vertices and every vertex of S is adjacent to at least k - 1 vertices in S. The minimum cardinality of a k-tuple dominating set of G is the k-tuple domination number of G. When k = 1, a k-tuple domination number is the well-studied domination...

Roman k-Tuple Domination in Graphs

For any integer $kgeq 1$ and any graph $G=(V,E)$ with minimum degree at least $k-1$‎, ‎we define a‎ ‎function $f:Vrightarrow {0,1,2}$ as a Roman $k$-tuple dominating‎ ‎function on $G$ if for any vertex $v$ with $f(v)=0$ there exist at least‎ ‎$k$ and for any vertex $v$ with $f(v)neq 0$ at least $k-1$ vertices in its neighborhood with $f(w)=2$‎. ‎The minimum weight of a Roman $k$-tuple dominatin...

On the Difference of Consecutive Primes

is greater than (c2/2)n. A simple calculation now shows that the primes satisfying (4) also satisfy the first inequality of (3) i΀ = e(ci) is chosen small enough. The second inequality of (3) is proved in the same way, which proves Theorem 2. Further, we obtain, as an immediate corollary of Theorem 1, that Received by the editors October 17, 1947. 1 P. Erdös and P. Turân, Some new questions on...