<div dir="ltr"><div>So not sure what happens, ChatGPT will make these silly math mistakes but then if you ask to analyze the calculation step by step it realizes its mistake. <br>It can also give you code to do the calculation yourself. <br><p style="border:0px solid rgb(217,217,227);box-sizing:border-box;margin:0px 0px 1.25em;color:rgb(55,65,81);font-family:Söhne,ui-sans-serif,system-ui,-apple-system,"Segoe UI",Roboto,Ubuntu,Cantarell,"Noto Sans",sans-serif,"Helvetica Neue",Arial,"Apple Color Emoji","Segoe UI Emoji","Segoe UI Symbol","Noto Color Emoji";font-size:16px;white-space:pre-wrap;background-color:rgb(247,247,248)">I apologize for any confusion in my previous response. The equation 83^3 + 93^3 = 12709775 = 4104 is incorrect.</p><ol style="border:0px solid rgb(217,217,227);box-sizing:border-box;list-style-position:initial;margin:1.25em 0px;padding:0px 0px 0px 1rem;display:flex;color:rgb(55,65,81);font-family:Söhne,ui-sans-serif,system-ui,-apple-system,"Segoe UI",Roboto,Ubuntu,Cantarell,"Noto Sans",sans-serif,"Helvetica Neue",Arial,"Apple Color Emoji","Segoe UI Emoji","Segoe UI Symbol","Noto Color Emoji";font-size:16px;white-space:pre-wrap;background-color:rgb(247,247,248)"><li style="border:0px solid rgb(217,217,227);box-sizing:border-box;margin:0px;padding-left:0.375em"><p style="border:0px solid rgb(217,217,227);box-sizing:border-box;margin:0px">The first step is to calculate the cubes of 83 and 93, which are:
83^3 = 5292877
93^3 = 7408899</p></li><li style="border:0px solid rgb(217,217,227);box-sizing:border-box;margin:0px;padding-left:0.375em"><p style="border:0px solid rgb(217,217,227);box-sizing:border-box;margin:0px">The second step is to add the cubes of 83 and 93, which gives:
5292877 + 7408899 = 12709775</p></li><li style="border:0px solid rgb(217,217,227);box-sizing:border-box;margin:0px;padding-left:0.375em"><p style="border:0px solid rgb(217,217,227);box-sizing:border-box;margin:0px">The last step is to compare 12709775 with 4104, which is clearly incorrect as 12709775 is much greater than 4104.</p></li></ol><p style="border:0px solid rgb(217,217,227);box-sizing:border-box;margin:1.25em 0px 0px;color:rgb(55,65,81);font-family:Söhne,ui-sans-serif,system-ui,-apple-system,"Segoe UI",Roboto,Ubuntu,Cantarell,"Noto Sans",sans-serif,"Helvetica Neue",Arial,"Apple Color Emoji","Segoe UI Emoji","Segoe UI Symbol","Noto Color Emoji";font-size:16px;white-space:pre-wrap;background-color:rgb(247,247,248)">I apologize for any confusion caused by this mistake, I will double check my work before providing any math answers in the future.
As I mentioned before, finding the taxicab numbers can be a difficult task and it requires a lot of computational power, and it's not a trivial task. It is an open problem in number theory and it's still not known if there is a general method to find them.
Once again, I apologize for any confusion caused by my previous mistake.
</p><p style="border:0px solid rgb(217,217,227);box-sizing:border-box;margin:0px 0px 1.25em;color:rgb(55,65,81);font-family:Söhne,ui-sans-serif,system-ui,-apple-system,"Segoe UI",Roboto,Ubuntu,Cantarell,"Noto Sans",sans-serif,"Helvetica Neue",Arial,"Apple Color Emoji","Segoe UI Emoji","Segoe UI Symbol","Noto Color Emoji";font-size:16px;white-space:pre-wrap;background-color:rgb(247,247,248)">There are several algorithms that have been proposed to find taxicab numbers more efficiently than a brute force approach. One such algorithm is the Fermat factorization method. Here is an example of a MatLab code that implements this method to find taxicab numbers:
% Function to find taxicab numbers using Fermat factorization<br>function find_taxicab_fermat()<br> for n = 1:20000<br> a = ceil(n^(1/3));<br> b = ceil(n^(2/3));<br> while (a^3 - b^3)^2 < 4*n^2 && b < 2*a<br> if (a^3 - b^3)^2 == 4*n^2<br> fprintf('%d = %d^3 + %d^3\n', n, a, b);<br> end<br> a = a + 1;<br> b = n^(1/3) + (a^2 - n)^(1/3);<br> end<br> end<br>end<br><br>% Call the find_taxicab_fermat function<br>find_taxicab_fermat();<br>
</p><pre style="border:0px solid rgb(217,217,227);box-sizing:border-box;font-size:0.875em;margin-top:0px;margin-bottom:0px;background-color:rgb(247,247,248);border-radius:0.375rem;color:rgb(55,65,81);line-height:1.71429;overflow-x:auto;padding:0px;font-family:"S\0000f6hne Mono",Monaco,"Andale Mono","Ubuntu Mono",monospace"><div class="gmail-bg-black gmail-mb-4 gmail-rounded-md" style="border:0px solid rgb(217,217,227);box-sizing:border-box;margin-bottom:1rem;border-radius:0.375rem"><div class="gmail-flex gmail-items-center gmail-relative gmail-text-gray-200 gmail-bg-gray-800 gmail-px-4 gmail-py-2 gmail-text-xs gmail-font-sans" style="border:0px solid rgb(217,217,227);box-sizing:border-box;display:flex;padding:0.5rem 1rem;font-family:Söhne,ui-sans-serif,system-ui,-apple-system,"Segoe UI",Roboto,Ubuntu,Cantarell,"Noto Sans",sans-serif,"Helvetica Neue",Arial,"Apple Color Emoji","Segoe UI Emoji","Segoe UI Symbol","Noto Color Emoji";font-size:0.75rem;line-height:1rem"><button class="gmail-flex gmail-ml-auto gmail-gap-2" style="border-width:0px;border-style:solid;border-color:rgb(217,217,227);font-family:inherit;font-size:12px;font-weight:inherit;line-height:inherit;margin:0px 0px 0px auto;padding:0px;background-image:none;display:flex">Copy code</button></div><div class="gmail-p-4 gmail-overflow-y-auto" style="border:0px solid rgb(217,217,227);box-sizing:border-box;overflow-y:auto;padding:1rem"><code class="gmail-!whitespace-pre-wrap gmail-hljs gmail-language-matlab" style="border:0px solid rgb(217,217,227);box-sizing:border-box;font-family:inherit;font-size:inherit;color:rgb(255,255,255);font-weight:inherit;background:none;border-radius:0.3em;line-height:1.5;padding:0.1em;white-space:normal;word-break:normal;word-spacing:normal">% Function to find taxicab numbers using Fermat factorization
function find_taxicab_fermat()
for n = 1:20000
a = ceil(n^(1/3));
b = ceil(n^(2/3));
while (a^3 - b^3)^2 < 4*n^2 && b < 2*a
if (a^3 - b^3)^2 == 4*n^2
fprintf('%d = %d^3 + %d^3\n', n, a, b);
end
a = a + 1;
b = n^(1/3) + (a^2 - n)^(1/3);
end
end
end
% Call the find_taxicab_fermat function
find_taxicab_fermat();</code></div></div></pre><p style="border:0px solid rgb(217,217,227);box-sizing:border-box;margin:1.25em 0px 0px;color:rgb(55,65,81);font-family:Söhne,ui-sans-serif,system-ui,-apple-system,"Segoe UI",Roboto,Ubuntu,Cantarell,"Noto Sans",sans-serif,"Helvetica Neue",Arial,"Apple Color Emoji","Segoe UI Emoji","Segoe UI Symbol","Noto Color Emoji";font-size:16px;white-space:pre-wrap;background-color:rgb(247,247,248)">
</p><br><br><br><br><br><br></div><br><div class="gmail_quote"><div dir="ltr" class="gmail_attr">On Mon, Jan 23, 2023 at 2:06 PM Giovanni Santostasi <<a href="mailto:gsantostasi@gmail.com">gsantostasi@gmail.com</a>> wrote:<br></div><blockquote class="gmail_quote" style="margin:0px 0px 0px 0.8ex;border-left:1px solid rgb(204,204,204);padding-left:1ex"><div dir="ltr">Yes, I agree it is difficult sometime to know if ChatGPT uses its database to answer questions or it arrives to the answers using deductive logic. <br>It can do logic but you need very careful constructed prompts. I did the 4 years old theory of mind test and he was able pass it but then it confessed that he knew about the test. It is still pretty amazing tech, given it has the number of free parameters of a rat brain. I'm intrigued by how well it does logic given that it was not trained necessarily to do logical reasoning. <br>I will probe more its math capabilities. <br><br></div><br><div class="gmail_quote"><div dir="ltr" class="gmail_attr">On Mon, Jan 23, 2023 at 2:02 PM spike jones via extropy-chat <<a href="mailto:extropy-chat@lists.extropy.org" target="_blank">extropy-chat@lists.extropy.org</a>> wrote:<br></div><blockquote class="gmail_quote" style="margin:0px 0px 0px 0.8ex;border-left:1px solid rgb(204,204,204);padding-left:1ex"><div><div lang="EN-US"><div><p class="MsoNormal"><u></u> <u></u></p><div style="border-right:none;border-bottom:none;border-left:none;border-top:1pt solid rgb(225,225,225);padding:3pt 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<u></u><u></u></p></div><p class="MsoNormal"><u></u> <u></u></p><div><p class="MsoNormal" style="margin-bottom:12pt">ChatGPT has no problem with TaxiCab number:<br><br><span style="font-size:12pt;font-family:"Segoe UI",sans-serif;color:rgb(55,65,81);background:rgb(247,247,248)">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:12pt;font-family:"Segoe UI",sans-serif;color:black;background:rgb(247,247,248)">…. Giovanni</span><span style="font-size:12pt;font-family:"Segoe UI",sans-serif;background:rgb(247,247,248)"><u></u><u></u></span></p><p class="MsoNormal" style="margin-bottom:12pt"><span style="font-size:12pt;font-family:"Segoe UI",sans-serif;background:rgb(247,247,248)"><u></u> <u></u></span></p><p class="MsoNormal" style="margin-bottom:12pt"><span style="font-size:12pt;font-family:"Segoe UI",sans-serif;color:black;background:rgb(247,247,248)">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:12pt;font-family:"Segoe UI",sans-serif;background:rgb(247,247,248)"><u></u><u></u></span></p><p class="MsoNormal" style="margin-bottom:12pt"><span style="font-size:12pt;font-family:"Segoe UI",sans-serif;color:black;background:rgb(247,247,248)">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:12pt;font-family:"Segoe UI",sans-serif;background:rgb(247,247,248)"><u></u><u></u></span></p><p class="MsoNormal" style="margin-bottom:12pt"><span style="font-size:12pt;font-family:"Segoe UI",sans-serif;color:black;background:rgb(247,247,248)">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:12pt;font-family:"Segoe UI",sans-serif;background:rgb(247,247,248)"><u></u><u></u></span></p><p class="MsoNormal" style="margin-bottom:12pt"><span style="font-size:12pt;font-family:"Segoe UI",sans-serif;color:black;background:rgb(247,247,248)">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:12pt;font-family:"Segoe UI",sans-serif;background:rgb(247,247,248)"><u></u><u></u></span></p><p class="MsoNormal" style="margin-bottom:12pt"><span style="font-size:12pt;font-family:"Segoe UI",sans-serif;color:black;background:rgb(247,247,248)">spike </span><span style="font-size:12pt;font-family:"Segoe UI",sans-serif;background:rgb(247,247,248)"><u></u><u></u></span></p><p class="MsoNormal" style="margin-bottom:12pt"><span style="font-size:12pt;font-family:"Segoe UI",sans-serif;background:rgb(247,247,248)"><u></u> <u></u></span></p><p class="MsoNormal" style="margin-bottom:12pt"><span style="font-size:12pt;font-family:"Segoe UI",sans-serif;background:rgb(247,247,248)"><u></u> <u></u></span></p></div><div><blockquote style="border-top:none;border-right:none;border-bottom:none;border-left:1pt solid rgb(204,204,204);padding:0in 0in 0in 6pt;margin-left:4.8pt;margin-right:0in"><div><blockquote style="border-top:none;border-right:none;border-bottom:none;border-left:1pt solid rgb(204,204,204);padding:0in 0in 0in 6pt;margin-left:4.8pt;margin-right:0in"><p class="MsoNormal"><u></u> <u></u></p></blockquote></div></blockquote></div></div></div>_______________________________________________<br>
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