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SUMMARY:Daniel Benjamin Flores (Purdue University)
DTSTART:20250523T190000Z
DTEND:20250523T192500Z
DTSTAMP:20260423T010421Z
UID:CANT2025/57
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/CANT2025/57/
 ">$K$-multimagic squares and magic squares of $k$th powers via the circle 
 method</a>\nby Daniel Benjamin Flores (Purdue University) as part of Combi
 natorial and additive number theory (CANT 2025)\n\nLecture held in CUNY Gr
 aduate Center - Science Center (4th floor).\n\nAbstract\nHere we investiga
 te $K$\\emph{-multimagic squares} of order $N$. These are $N \\times N$ ma
 gic squares which remain magic after raising each element to the $k$th pow
 er for all $2 \\le k \\le K$. Given $K \\ge 2$\, we consider the problem o
 f establishing the smallest integer $N_0(K)$ for which there exist \\emph{
 nontrivial} $K$-multimagic squares of order $N_0(K)$. \n\nPrevious results
  on multimagic squares show that $N_0(K) \\le (4K-2)^K$ for large $K$. We 
 use the Hardy-Littlewood circle method to improve this to \n\\[N_0(K) \\le
  2K(K+1)+1.\\]\nThe intricate structure of the coefficient matrix poses si
 gnificant technical challenges for the circle method. We overcome these ob
 stacles by generalizing the class of Diophantine systems amenable to the c
 ircle method and demonstrating that the multimagic square system belongs t
 o this class for all $N \\ge 4$. We additionally establish the existence o
 f infinitely many $N \\times N$ magic squares of distinct $k$th powers as 
 soon as\n\\[N > 2\\min\\{2^k\,\\lceil k(\\log k +4.20032) \\rceil \\}.\\]\
 nThis result marks progress toward resolving an open problem popularized b
 y Martin Gardner in 1996\, which asks whether a $3 \\times 3$ magic square
  of distinct squares exists.\n
LOCATION:https://researchseminars.org/talk/CANT2025/57/
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