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</o:shapelayout></xml><![endif]--></head><body lang=EN-US link=blue vlink=purple style='word-wrap:break-word'><div class=WordSection1><p class=MsoNormal><o:p> </o:p></p><div style='border:none;border-top:solid #E1E1E1 1.0pt;padding:3.0pt 0in 0in 0in'><p class=MsoNormal><b>.</b>> <b>On Behalf Of </b>Giovanni Santostasi via extropy-chat<br><b>Subject:</b> Re: [ExI] Google LaMDA avheives enlightenment<o:p></o:p></p></div><p class=MsoNormal><o:p> </o:p></p><div><p class=MsoNormal style='margin-bottom:12.0pt'>ChatGPT has no problem with TaxiCab number:<br><br><span style='font-size:12.0pt;font-family:"Segoe UI",sans-serif;color:#374151;background:#F7F7F8'>The smallest number that can be expressed as the sum of two cubes in three different ways is 1729. It can be expressed as the sum of the cubes of 12 and 1 (12^3 + 1^3 = 1729), as well as the cubes of 9 and 10 (9^3 + 10^3 = 1729), and as the cubes of 1 and 12 (1^3 + 12^3 = 1729). This is known as the "taxicab number" and is the smallest such number that can be expressed in multiple ways</span><span style='font-size:12.0pt;font-family:"Segoe UI",sans-serif;color:black;background:#F7F7F8'>…. Giovanni</span><span style='font-size:12.0pt;font-family:"Segoe UI",sans-serif;background:#F7F7F8'><o:p></o:p></span></p><p class=MsoNormal style='margin-bottom:12.0pt'><span style='font-size:12.0pt;font-family:"Segoe UI",sans-serif;background:#F7F7F8'><o:p> </o:p></span></p><p class=MsoNormal style='margin-bottom:12.0pt'><span style='font-size:12.0pt;font-family:"Segoe UI",sans-serif;color:black;background:#F7F7F8'>Ja Giovanni but of course there is plenty of online documentation for taxicab numbers because of the famous Ramanujan anecdote.</span><span style='font-size:12.0pt;font-family:"Segoe UI",sans-serif;background:#F7F7F8'><o:p></o:p></span></p><p class=MsoNormal style='margin-bottom:12.0pt'><span style='font-size:12.0pt;font-family:"Segoe UI",sans-serif;color:black;background:#F7F7F8'>To find a number which we can call triple taxis, we need a little more sophisticated software. I can offer 8 of them, but I urge you to spin up some code or search online for them first.</span><span style='font-size:12.0pt;font-family:"Segoe UI",sans-serif;background:#F7F7F8'><o:p></o:p></span></p><p class=MsoNormal style='margin-bottom:12.0pt'><span style='font-size:12.0pt;font-family:"Segoe UI",sans-serif;color:black;background:#F7F7F8'>This algorithm I derived can get us arbitrarily many triple taxis in a reasonable time (a few minutes (but the code is too large for the margin of this post.)</span><span style='font-size:12.0pt;font-family:"Segoe UI",sans-serif;background:#F7F7F8'><o:p></o:p></span></p><p class=MsoNormal style='margin-bottom:12.0pt'><span style='font-size:12.0pt;font-family:"Segoe UI",sans-serif;color:black;background:#F7F7F8'>If no one can find any triple taxis, either by code or by search, I will post the solutions on my blog, Math Geezer, which needs some content.</span><span style='font-size:12.0pt;font-family:"Segoe UI",sans-serif;background:#F7F7F8'><o:p></o:p></span></p><p class=MsoNormal style='margin-bottom:12.0pt'><span style='font-size:12.0pt;font-family:"Segoe UI",sans-serif;color:black;background:#F7F7F8'>spike </span><span style='font-size:12.0pt;font-family:"Segoe UI",sans-serif;background:#F7F7F8'><o:p></o:p></span></p><p class=MsoNormal style='margin-bottom:12.0pt'><span style='font-size:12.0pt;font-family:"Segoe UI",sans-serif;background:#F7F7F8'><o:p> </o:p></span></p><p class=MsoNormal style='margin-bottom:12.0pt'><span style='font-size:12.0pt;font-family:"Segoe UI",sans-serif;background:#F7F7F8'><o:p> </o:p></span></p></div><div><blockquote style='border:none;border-left:solid #CCCCCC 1.0pt;padding:0in 0in 0in 6.0pt;margin-left:4.8pt;margin-right:0in'><div><blockquote style='border:none;border-left:solid #CCCCCC 1.0pt;padding:0in 0in 0in 6.0pt;margin-left:4.8pt;margin-right:0in'><p class=MsoNormal><o:p> </o:p></p></blockquote></div></blockquote></div></div></body></html>