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URL:https://uwaterloo.ca/pure-mathematics/events/number-theory-seminar-109
SUMMARY:Number Theory Seminar
CLASS:PUBLIC
DESCRIPTION:PIERRE POPOLI\, UNIVERSITY OF LIÈGE\n\n\"ON THE BINARY DIGITS
  OF $N$ AND $N^2$\"\n\nLet $s(n)$ denote the sum of digits in the binary e
 xpansion of the\ninteger $n.$ Hare\, Laishram and Stoll (2011) studied the
  number of odd\nintegers such that $s(n)=s(n^2)=k\,$ for a given positive 
 integer $k.$\nThe remaining cases that could not be treated by theses auth
 ors were\n$k=9\,$ 10\, 11\, 14 or 15. In this talk\, I will present the re
 sults of\nour article on the cases $k=9\,$10 and 11 and the difficulties t
 o\nsettle for the two remaining cases $k=14$ and 15. \n\nA related proble
 m is to study perfect squares of odd integers with\nfour binary digits. Be
 nnett\, Bugeaud and Mignotte (2012) proved that\nthere are only finitely m
 any solutions and conjectured that the set of\nsolutions is composed of 13
 \, 15\, 47 and 111. In the same paper\, we\ngive an algorithm to find all 
 solutions with fixed sum of digits\nvalue\, supporting this conjecture\, a
 s well as show related results for\nperfect squares of odd integers with f
 ive binary digits. \nThis is joint work with Aloui\, Jamet\, Kaneko\, Kop
 ecki and Stoll.\n \n\nZoom\nlink: https://uwaterloo.zoom.us/j/9473021078
 7?pwd=NUEwNk8rZlFqUm9vSDZWY1lCL1IyZz09
DTSTAMP:20260605T022051Z
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