BEGIN:VCALENDAR VERSION:2.0 PRODID:researchseminars.org CALSCALE:GREGORIAN X-WR-CALNAME:researchseminars.org BEGIN:VEVENT SUMMARY:Carl Pomerance (Dartmouth College) DTSTART:20260714T173000Z DTEND:20260714T182000Z DTSTAMP:20260710T111708Z UID:CANT2026/26 DESCRIPTION:Title: Two topics in combinatorial number theory\nby Carl Pomerance (Dartmo uth College) as part of Combinatorial and additive number theory seminar ( CANT 2026)\n\nLecture held in Science Center in the CUNY Graduate Center ( 4th floor).\n\nAbstract\nThe first topic: In a paper with Erd\\H os from 4 0 years ago\,\nwe considered the set of residues $a \\bmod n$ where\n$a^{n -1} \\equiv 1 \\pmod n$.\nIf $n$ is composite\, these are the bases for wh ich $n$ is a pseudoprime.\nRecently\, Lenstra asked me about the set of re sidues $a \\bmod n$\nwhere $a^n \\equiv 1 \\pmod n$\, which is related to a problem he is\nworking on about conditions that ensure a ring is commuta tive.\nSome of the methods from the old paper were useful in the new\nprob lem\, but not all. I will discuss the more general problem of subgroups of the multiplicative group mod $n$. The second topic: I will discuss\nsome old and new problems on coprime matchings: These are perfect\nmatchings be tween two equally numerous sets of integers\, where each matched pair is r elatively prime. Some examples: Given two intervals\nof $n$ consecutive in tegers is there a coprime matching between them?\nIf both intervals are $\ \{1\,2\,\\dots\,n\\}$\, how many such matchings are\nthere? For a positive integer $n$\, is there a coprime matching between\nthe set $D(n)$ of divi sors of $n$ and an interval of $D(n)$ consecutive\nintegers? This last pro blem reflects joint work with Nathan McNew.\n LOCATION:https://researchseminars.org/talk/CANT2026/26/ END:VEVENT END:VCALENDAR