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BEGIN:VEVENT
UID:f1fe5f60f31e6b493b13e210305b1189
CATEGORIES:Experimental Mathematics Seminar
CREATED:20211028T140406
SUMMARY:Lucas congruences modulo p2
LOCATION:Zoom
DESCRIPTION:Abstract: In the 1870s, Lucas obtained a beautiful formula for binomial coe
 fficients modulo p. Namely, Binomial[n, m] is congruent modulo p to the pro
 duct of the binomial coefficients whose arguments are the base-p digits of 
 n and m, taken pairwise. Variations and generalizations of this formula hav
 e been actively investigated since. In particular, for which n and m does L
 ucas' congruence hold not just modulo p but modulo p2? The answer is relate
 d to some hidden rotational symmetry in Pascal's triangle. A similar result
  holds for the Apéry numbers, which Gessel showed satisfy a Lucas congruen
 ce in 1982\n
X-ALT-DESC;FMTTYPE=text/html:<p><em style="color: #000000; font-family: 'Times New Roman'; font-size: me
 dium; font-weight: 400; letter-spacing: normal; orphans: 2; text-align: sta
 rt; text-indent: 0px; text-transform: none; white-space: normal; widows: 2;
  word-spacing: 0px;">Abstract</em>: In the 1870s, Lucas obtained a beautifu
 l formula for binomial coefficients modulo p. Namely, Binomial[n, m] is con
 gruent modulo p to the product of the binomial coefficients whose arguments
  are the base-p digits of n and m, taken pairwise. Variations and generaliz
 ations of this formula have been actively investigated since. In particular
 , for which n and m does Lucas' congruence hold not just modulo p but modul
 o p<sup style="color: #000000; font-family: 'Times New Roman'; font-weight:
  400; letter-spacing: normal; orphans: 2; text-align: start; text-indent: 0
 px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px
 ;">2</sup>? The answer is related to some hidden rotational symmetry in Pas
 cal's triangle. A similar result holds for the Apéry numbers, which Gessel
  showed satisfy a Lucas congruence in 1982</p>
CONTACT:Eric Rowland, Hofstra University
X-EXTRAINFO:Zoom link: https://rutgers.zoom.us/j/94346444480\npassword: The 20th Catala
 n number, alias (40)!/(20!*21!), alias 6564120420
DTSTAMP:20260827T121312
DTSTART;TZID=America/New_York:20211104T170000
DTEND;TZID=America/New_York:20211104T180000
SEQUENCE:0
TRANSP:OPAQUE
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