Quintasymptotic primes and four results of Schenzel

Quintasymptotic sequences over an ideal and quintasymptotic cograde

‎Let $I$ denote an ideal of a Noetherian ring $R$‎. ‎The purpose of‎ ‎this article is to introduce the concepts of quintasymptotic‎ ‎sequences over $I$ and quintasymptotic cograde of $I$‎, ‎and to show that they play a role analogous to quintessential sequences‎ ‎over $I$ and quintessential cograde of $I$‎. ‎We show that‎, ‎if $R$ is‎ ‎local‎, ‎then the quintasymptotic cograde of $I$ is una...

Finding Four Million Large Random Primes

Validation markers in introduction and results and discussion sections of research articles from four disciplines

In the last three decades, genre of research article, among other genres of academic writings, hasreceived the greatest attention. This attention is due to the vital role that research article plays inthe legitimating of claims and disciplines. Johns and Swales (2002) relate this attention to theintensive review process that research article goes through before getting “valorized and ratifiedby...

Circular Units of Real Abelian Fields with Four Ramified Primes

Abstract. In this paper we study the groups of circular numbers and circular units in Sinnott’s sense in real abelian fields with exactly four ramified primes under certain conditions. More specifically, we construct Z-bases for them in five special infinite families of cases. We also derive some results about the corresponding module of relations (in one family of cases, we show that the modul...

NFS with Four Large Primes: An Explosive Experiment

The purpose of this paper is to report the unexpected results that we obtained while experimenting with the multi-large prime variation of the general number field sieve integer factoring algorithm (NFS, cf. [8]). For traditional factoring algorithms that make use of at most two large primes, the completion time can quite accurately be predicted by extrapolating an almost quartic and entirely ‘...