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DTSTAMP:20260410T082000Z
CREATED:20220120T082542Z
UID:ATEvent-0942d35f122149e99619e4e11b3450ea
LAST-MODIFIED:20220406T110842Z
SUMMARY:RISC Forum
DTSTART:20220425T113000Z
DTEND:20220425T114500Z
DESCRIPTION:Prof. Armin Straub: Lucas congruences and congruence schem
 es.\n\n\nAbstract: It is a well-known and beautiful classical result o
 f Lucas that\, modulo a prime $p$\, the binomial coefficients satisfy 
 the congruences\n\begin{equation*}\n  \binom{n}{k} \equiv \binom{n_0}{
 k_0} \binom{n_1}{k_1} \cdots \binom{n_r}{k_r},\n\end{equation*}\nwhere
  $n_i$\, respectively $k_i$\, are the $p$-adic digits of $n$ and $k$. 
  Many interesting integer sequences have been shown to satisfy version
 s of these congruences.  For instance\, Gessel has done so for the num
 bers used by Ap\'ery in his proof of the irrationality of $zeta(3)$.  
 We make the observation that a sequence satisfies Lucas congruences mo
 dulo $p$ if and only if its values modulo $p$ can be described by a li
 near $p$-scheme\, as introduced by Rowland and Zeilberger\, with a sin
 gle state.  This simple observation suggests natural generalizations o
 f the notion of Lucas congruences.  To illustrate this point\, we deri
 ve explicit generalized Lucas congruences for integer sequences that c
 an be represented as certain constant terms.  This talk includes joint
  work with Joel Henningsen.
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