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DTSTART:20240310T070000
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DTSTART:20241103T060000
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DTSTART;TZID=America/Toronto:20241119T102000
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URL:https://uwaterloo.ca/pure-mathematics/events/number-theory-seminar-133
SUMMARY:Number Theory Seminar
CLASS:PUBLIC
DESCRIPTION:TANLEY XIAO\, UNIVERSITY OF NORTHERN BRITISH COLUMBIA \n\nOn Bu
 chi's problem \n\nIn 1970\, J. Richard Buchi showed that there is no gener
 al algorithm\nwhich decides whether a general quadratic equation in arbitr
 arily many\nvariables has a solution in the integers\, subject to a hypoth
 esis\nwhich would be named Buchi's Problem. Buchi's result is a\nstrengthe
 ning of the negative answer of Hilbert's Tenth Problem. \n\nBuchi's proble
 m is an elegant number theoretic problem in its own\nright. It asserts tha
 t there exists a positive integer M such that\nwhenever a finite sequence 
 x_0^2\, x_1^2\,...\, x_n^2 of increasing\nsquare integers has constant sec
 ond difference equal to 2 (that is\,\nx_{j+2}^2 - 2 x_{j+1}^2 + x_j^2 = 2 
 for j = 0\, ...\, n-2)\, then either\nn \\leq M or x_j^2 = (x_0 + j)^2 for
  j = 1\, ...\, n. \n\nIn this talk\,  we show that Buchi's problem has an
  affirmative answer\nwith M = 5. In other words\, there are no non-trivial
  quintuple of\nincreasing square integers with constant difference equal t
 o 2. \n\nMC 5479
DTSTAMP:20260408T022754Z
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