The limits of Buchstab's iteration sieve

AbstractAn analysis is presented leading to explicit equations for the limits of the Buchstab iteration sieve. Moreover, the limits are computed for some values of the relevant parameter κ

Amicable pairs and aliquot cycles for elliptic curves

An amicable pair for an elliptic curve E/Q is a pair of primes (p,q) of good reduction for E satisfying #E(F_p) = q and #E(F_q) = p. In this paper we study elliptic amicable pairs and analogously defined longer elliptic aliquot cycles. We show that there exist elliptic curves with arbitrarily long aliqout cycles, but that CM elliptic curves (with j not 0) have no aliqout cycles of length greater than two. We give conjectural formulas for the frequency of amicable pairs. For CM curves, the derivation of precise conjectural formulas involves a detailed analysis of the values of the Grossencharacter evaluated at a prime ideal P in End(E) having the property that #E(F_P) is prime. This is especially intricate for the family of curves with j = 0.Comment: 53 page

Suspected Lynch syndrome associated MSH6 variants: A functional assay to determine their pathogenicity

Lynch syndrome (LS) is a hereditary cancer predisposition caused by inactivating mutations in DNA mismatch repair (MMR) genes. Mutations in the MSH6 DNA MMR gene account for approximately 18% of LS cases. Many LS-associated sequence variants are nonsense and frameshift mutations that clearly abrogate MMR activity. However, missense mutations whose functional implications are unclear are also frequently seen in suspected-LS patients. To conclusively diagnose LS and enroll patients in appropriate surveillance programs to reduce morbidity as well as mortality, the functional consequences of these variants of uncertain clinical significance (VUS) must be defined. We present an oligonucleotide-directed mutagenesis screen for the identification of pathogenic MSH6 VUS. In the screen, the MSH6 variant of interest is introduced into mouse embryonic stem cells by site-directed mutagenesis. Subsequent selection for MMR-deficient cells using the DNA damaging agent 6-thioguanine (6TG) allows the identification of MMR abrogating VUS because solely MMR-deficient cells survive 6TG exposure. We demonstrate the efficacy of the genetic screen, investigate the phenotype of 26 MSH6 VUS and compare our screening results to clinical data from suspected-LS patients carrying these variant alleles