Let j(n) denote the smallest positive integer m such that every sequence of m consecutive integers contains an integer prime to n. Let P n be the product of the first n primes and define h(n) = j(P n ). Presently, h(n) is only known for n ≤ 24. In this paper, we describe an algorithm that enabled the calculation of h(n) for n < 50.
The method of invariants is an important approach in biology for determining phylogenetic information which avoids the problems involving long branch lengths that plague other methods. In this paper, we verify the conjecture on the number of algebraic generators for the ideal of polynomial invariants. We also study rational and analytic invariants and prove several criteria concerning when it suffices to work with polynomial invariants.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.