<div dir="ltr"><div dir="ltr"><div class="gmail_default" style="font-family:arial,helvetica,sans-serif"><span style="font-family:Arial,Helvetica,sans-serif">On Mon, Apr 13, 2026 at 12:14 PM Adrian Tymes via extropy-chat <<a href="mailto:extropy-chat@lists.extropy.org">extropy-chat@lists.extropy.org</a>> wrote:</span></div></div><div class="gmail_quote gmail_quote_container"><div dir="ltr" class="gmail_attr"><br></div><blockquote class="gmail_quote" style="margin:0px 0px 0px 0.8ex;border-left:1px solid rgb(204,204,204);padding-left:1ex"><font size="4" face="georgia, serif"><i><span class="gmail_default" style="font-family:arial,helvetica,sans-serif">> </span>John's claim that "there is no shortcut" for computing the Goldbach<br>conjecture at high numbers is false - again, in practice.  Ask your<br>favorite AI, "When proving the Goodbach conjecture works up to some<br>staggeringly high number, in practice are there optimizations one can<br>take to prove this in a plausible amount of time?", and it'll probably<br>tell you.</i></font></blockquote><div><br></div><div><font size="4" face="tahoma, sans-serif"><b>I Took your advice and<span class="gmail_default" style=""> </span>asked Gemini that very question and it did come up with <span class="gmail_default" style="">some </span>very clever<span class="gmail_default" style=""> </span>optimization techniques that would dramatically speed things <span class="gmail_default" style="">up</span>, Gemini concluded with this:</b></font></div><div><br></div><div dir="ltr" class="gmail_attr"><font size="4"><b><span class="gmail-citation-8 gmail-citation-end-8" style="font-family:"Google Sans Text",sans-serif;line-height:1.15;margin-top:0px"><span class="gmail_default" style="font-family:arial,helvetica,sans-serif">"</span>As of the latest major computations, the conjecture has been verified to </span><span class="gmail-math-inline" style="font-family:"Google Sans Text",sans-serif;line-height:1.15;margin-top:0px">4 <span class="gmail_default" style="font-family:arial,helvetica,sans-serif">*</span>10^18<span class="gmail_default" style="font-family:arial,helvetica,sans-serif">.</span></span> <span class="gmail-citation-6 gmail-citation-end-6" style="font-family:"Google Sans Text",sans-serif;line-height:1.15;margin-top:0px">To give you a sense of the efficiency: on a standard modern processor, testing an interval of </span><span class="gmail-math-inline" style="font-family:"Google Sans Text",sans-serif;line-height:1.15;margin-top:0px">10^12</span><span class="gmail-citation-5" style="font-family:"Google Sans Text",sans-serif;line-height:1.15;margin-top:0px"> (one trillion) integers near the current limit takes roughly </span><span style="font-family:"Google Sans Text",sans-serif;line-height:1.15;margin-top:0px"><span class="gmail-citation-5" style="line-height:1.15;margin-top:0px">45 to 50 minutes</span></span><span class="gmail-citation-5 gmail-citation-end-5" style="font-family:"Google Sans Text",sans-serif;line-height:1.15;margin-top:0px">.<span style="line-height:1.15;margin-top:0px"><sup class="gmail-superscript" style="line-height:1.15;margin-top:0px;background-color:transparent"></sup></span></span> Without these optimizations, it would take centuries.<span class="gmail_default" style="font-family:arial,helvetica,sans-serif">"</span></b></font></div><div dir="ltr" class="gmail_attr"><span class="gmail_default" style="font-family:arial,helvetica,sans-serif"><br></span></div><div dir="ltr" class="gmail_attr"><font size="4" face="tahoma, sans-serif"><b>And that sounds great but even with these<span class="gmail_default" style=""> dramatic speed ups, to go from </span>4<span class="gmail_default" style="">*</span>10^18<span class="gmail_default" style=""> to </span><span class="gmail_default" style="">5</span><span class="gmail_default" style="">*</span>10^18<span class="gmail_default" style=""> would take 34,722 days, or 95 years. And as you proceed to deal with larger and larger numbers the slower your progress will be.  </span></b></font></div><div dir="ltr" class="gmail_attr"><font size="4" face="tahoma, sans-serif"><b><span class="gmail_default" style=""><br></span></b></font></div><div dir="ltr" class="gmail_attr"><font size="4"><font face="tahoma, sans-serif"><b><span class="gmail_default" style="">And if you were trying to calculate Busy Beaver numbers things would be even worse because at some point BB grows faster been super exponential, in fact it grows <u>faster than ANY conceivable computable function</u>. If there was a computable function that grew faster than </span></b></font><b style="font-family:tahoma,sans-serif"><span class="gmail_default" style="">Busy Beaver then you could use it to solve the Halting Problem, but Alan Turing proved in 1936 that was logically impossible. </span></b></font></div><div dir="ltr" class="gmail_attr"><b style="font-family:tahoma,sans-serif;font-size:large"><span class="gmail_default"><br></span></b></div><div class="gmail_attr"><b style="font-family:tahoma,sans-serif;font-size:large"><span class="gmail_default">  John K Clark</span></b></div><div dir="ltr" class="gmail_attr"><b style="font-family:tahoma,sans-serif;font-size:large"><span class="gmail_default"><br></span></b></div><blockquote class="gmail_quote" style="margin:0px 0px 0px 0.8ex;border-left:1px solid rgb(204,204,204);padding-left:1ex"><br>
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