Abstract. Given a set of integers with no three in arithmetic progression, we construct a Stanley sequence by adding integers greedily so that no arithmetic progression is formed. This paper offers two main contributions to the theory of Stanley sequences. First, we characterize well-structured Stanley sequences as solutions to constraints in modular arithmetic, defining the modular Stanley sequences. Second, we introduce the basic Stanley sequences, where elements arise as the sums of subsets of a basis sequence, which in the simplest case is the powers of 3. Applications of our results include the construction of Stanley sequences with arbitrarily large gaps between terms, answering a weak version of a problem by Erdős et al. Finally, we generalize many results about Stanley sequences to p-free sequences, where p is any odd prime.
Given a finite set of nonnegative integers A with no three-term arithmetic progressions, the Stanley sequence generated by A, denoted as S(A), is the infinite set created by beginning with A and then greedily including strictly larger integers which do not introduce a three-term arithmetic progression in S
Complement-fixing antibodies to a herpes-like virus derived from a Burkitt tumor-cell line developed in each of 21 patients with infectious mononu-cleosis. These antibodies were absent in all serums before the patients became ill, appeared during the early phases of illness, and persisted for long periods of time. These antibodies are distinct from heterophile antibodies. None of the patients developed immune responses to herpes simplex, cytomegalo-, or reoviruses in the course of their illness. The data suggest that the development of complement-fixing antibodies to this herpes-like virus in these patients may be linked to infectious mononucleosis.
In this note, we prove that there exists a classical Hilbert modular cusp form over Q( √ 5) of partial weight one which does not arise from the induction of a Grössencharacter from a CM extension of Q( √ 5).We originally learnt of this problem through Fred Diamond. In conversations with Kevin Buzzard, Don Blasius, and Fred Diamond, it became clear that the question of whether such forms existed was apparent to the authors of [1] in the '80s (and may well have occurred to others before then). The question gained some urgency with the advent of Fraser Jarvis' construction of Galois representations for partial weight one forms [6] in the mid-'90s, since, if the only such forms were CM, then [6] would be a trivial consequence of Class field theory. We have heard several reports of the question being raised again at this time. In light of these stories, we feel safe in calling the problem a "well-known folklore question."
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.