We have constructed a Carmichael number with 10,333,229,505 prime factors, and have also constructed Carmichael numbers with k prime factors for every k between 3 and 19,565,220. These computations are the product of implementations of two new algorithms for the subset product problem that exploit the non-uniform distribution of primes p with the property that p − 1 divides a highly composite Λ.

We exhibit a simplified version of the construction of a field of Morley rank p with a predicate of rank p − 1, extracting the main ideas for the construction from previous papers and refining the arguments. Moreover, an explicit axiomatization is given, and ranks are computed.

A subset D of the vertex set of a graph G is a (k, p)-dominating set if every vertex v ∈ V (G) \ D is within distance k to at least p vertices in D. The parameter γk,p(G) denotes the minimum cardinality of a (k, p)-dominating set of G. In 1994, Bean, Henning and Swart posed the conjecture that γk,p(G) ≤ p p+k n(G) for any graph G with δk(G) ≥ k + p − 1, where the latter means that every vertex ...

For a finite group G, let cd(G) be the set of irreducible complex character degrees of G forgetting multiplicities and X1(G) be the set of all irreducible complex character degrees of G counting multiplicities. Suppose that p is a prime number. We prove that if G is a finite group such that |G| = |PGL(2, p)|, p ∈ cd(G) and max(cd(G)) = p+1, then G ∼= PGL(2, p), SL(2, p) or PSL(2, p) × A, where ...

Abstract. We consider the steady states of the Gierer–Meinhardt system on all of R: εΔa − a + a hq = 0, Δh − h + a hs = 0 with an additional restriction q = p − 1. In the limit ε → 0, we use formal asymptotics to construct a solution whose activator component a concentrates on a circle. Under the additional constraints p > 1, m > 0, and 1 < m− s < 3, we find that such a solution exists and is u...

The post-high school transition period is believed to be associated with considerable changes in social networks, yet longitudinal studies documenting these changes are scarce. To address this gap, the current research explored 3 relevant issues. First, changes in participants' relationship characteristics during the transition from high school were examined. Second, the roles of personality tr...

BACKGROUND Research has reported that pregnant women and mothers become forgetful. However, in these studies, women are not recruited prior to pregnancy, samples are not representative and studies are underpowered. AIMS The current study sought to determine whether pregnancy and motherhood are associated with brief or long-term cognitive deterioration using a representative sample and measuri...