The search for first occurrences of prime gaps and maximal prime gaps is extended to 1015. New maximal prime gaps of 806 and 906 are found, and sixty-two previously unpublished first occurrences are found for gaps varying from 676 to 906.

For any integer n ≥ 1, let j(n) denote the Jacobsthal function, and ω(n) the number of distinct prime divisors of n. In 1962 Jacobsthal conjectured that for any integer r ≥ 1, the maximal value of j(n) when n varies over N with ω(n) = r is attained when n is the product of the first r primes. We show that this is true for r ≤ 23 and fails at r = 24, thus disproving Jacobsthal’s conjecture.

Consider multisets A in the group G = (Z/nZ) such that no non-empty subset has sum zero. It is known for long that the maximal cardinality of such a set is 2n − 2, and there is a conjecture of Gao and Geroldinger describing the structure of such sets of maximal cardinality (called “property B”). Recently, Gao, Geroldinger and Grynkiewicz showed that it is enough to prove the conjecture for n pr...

Abstract In this paper, we call a finite group G G an N L M NLM -group ( C NCM -group, respectively) if every non-normal cyclic subgroup of prime order or 4 (prime power order, in is contained maximal . Using the property -groups and -groups, give new necessary suffic...

1. I n t r o d u c t i o n In 1640, Pierre de Fermat conjectured that all numbers F m = 2 2 ' " + 1 for m = 0, 1, 2 , . . . , (1) are prime, which was later found to be incorrect. The numbers Fm are called Fermat numbers after him. If Fm is prime, we say that it is a Fermat prime. Until the end of the 18th century, Fermat numbers were most likely a mathe­ matical curiosity. The interest in the ...

It is proved that, for a prime number p, showing that an n-element matroid is not representable over GF (p) requires only O(n) rank evaluations.

suppose $g$ is a split connected‎ ‎reductive orthogonal or symplectic group over an infinite field‎ ‎$f,$ $p=mn$ is a maximal parabolic subgroup of $g,$ $frak{n}$ is‎ ‎the lie algebra of the unipotent radical $n.$ under the adjoint‎ ‎action of its stabilizer in $m,$ every maximal prehomogeneous‎ ‎subspaces of $frak{n}$ is determined‎.

Let S be a semigroup and K be a field. A K-space K[S], with basis S and with multiplication extending, in a natural way, the operation on S, is called a semigroup algebra. It remains an open problem to characterize semigroup algebras that are a prime Noetherian maximal order. In this thesis, we give an answer to the problem for a large class of cancellative semigroups and we illustrate these re...

The maximal generic words problem was proposed by Kucherov et al. (SPIRE 2012). Let D be a set of documents. In this problem, given a pattern P and a threshold d ≤ |D|, we want to compute all right-maximal extensions of P which occur in at least d distinct documents. They proposed an O(n)-space data structure which can solve this problem in O(|P |+ rocc) time where n is the total length of docu...

An algorithm is presented which, given a positive integer n, will either factor n or prove it to be prime. The algorithm takes O(«) steps.