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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>On Wed, Mar 24, 2021 at 9:39 AM William Flynn Wallace via extropy-chat <<a href="mailto:extropy-chat@lists.extropy.org">extropy-chat@lists.extropy.org</a>> wrote:<o:p></o:p></p><p class=MsoNormal><span style='font-size:18.0pt;font-family:"Comic Sans MS";color:black'>He wanted to know the meaning of this statement:<o:p></o:p></span></p><p class=MsoNormal><span style='font-size:18.0pt;font-family:"Comic Sans MS";color:black'><o:p> </o:p></span></p><p class=MsoNormal><span style='font-family:"Arial",sans-serif;color:black'>Mathematics is true, but it doesn't exist.</span><span style='font-size:10.0pt;font-family:"Arial",sans-serif;color:black'>"</span><span style='font-family:"Comic Sans MS";color:black'><o:p></o:p></span></p><p class=MsoNormal><span style='font-family:"Comic Sans MS";color:black'><o:p> </o:p></span></p><p class=MsoNormal><span style='font-size:10.0pt;font-family:"Arial",sans-serif;color:black'>bill w</span><span style='font-family:"Comic Sans MS";color:black'><o:p></o:p></span></p><p class=MsoNormal><o:p> </o:p></p><p class=MsoNormal><o:p> </o:p></p><p class=MsoNormal><span style='font-size:16.0pt'>Consider the conjecture “An odd perfect number exists.”<o:p></o:p></span></p><p class=MsoNormal><span style='font-size:16.0pt'><o:p> </o:p></span></p><p class=MsoNormal><span style='font-size:16.0pt'>That one has always intrigued me. About a quarter of the even numbers are abundant, and of those only 47 of the first 10^(25956377) numbers is perfect. That might not be perfectly clear because we cannot imagine numbers that have 26 million decimal digits. Of those numbers with 26 million decimal digits, only 47 are perfect.<o:p></o:p></span></p><p class=MsoNormal><span style='font-size:16.0pt'><o:p> </o:p></span></p><p class=MsoNormal><span style='font-size:16.0pt'><o:p> </o:p></span></p><p class=MsoNormal><span style='font-size:16.0pt'>But only about a fifth of a percent of odd numbers are abundant. So for every odd abundant number there are about 125 even ones.<o:p></o:p></span></p><p class=MsoNormal><span style='font-size:16.0pt'><o:p> </o:p></span></p><p class=MsoNormal><span style='font-size:16.0pt'>OK. So if we figured out that 47 out of (a number with 26 million digits) is perfect, we can say that one out of about (well, still about 26 million digits) abundant numbers is perfect, because a quarter of 26 million digit number is still a 26 million digit number. It is big enough that it doesn’t notice it was divided by 4.<o:p></o:p></span></p><p class=MsoNormal><span style='font-size:16.0pt'><o:p> </o:p></span></p><p class=MsoNormal><span style='font-size:16.0pt'>If perfect numbers are that rare, we have no way of knowing if an odd perfect number exists. We would expect only about .4 odd perfect numbers (47)/(125) and that 40% chance of an odd perfect is hiding somewhere in the first 26 million digit numbers. So we have no known way of finding it.<o:p></o:p></span></p><p class=MsoNormal><span style='font-size:16.0pt'><o:p> </o:p></span></p><p class=MsoNormal><span style='font-size:16.0pt'>We don’t know if it exists or does not exist. We know it has to be greater than a 300 digit number but that really doesn’t tell us much because the fraction of numbers 26 million digits or less that has less than 300 digits is negligible. 10^26 million is big enough that it doesn’t notice if it is divided by a mere 10^300, a mere googol*googolplex. This denominator of a googol*googolplex is swallowed, completely dominated by that numerator.<o:p></o:p></span></p><p class=MsoNormal><span style='font-size:16.0pt'><o:p> </o:p></span></p><p class=MsoNormal><span style='font-size:16.0pt'>I think there is an odd perfect number, and if so, there are infinitely many of them (because you can establish an asymptotic ratio and multiply it up as faaaaaar as you want to imagine. Hey, that’s what imaginations are for.<o:p></o:p></span></p><p class=MsoNormal><span style='font-size:16.0pt'><o:p> </o:p></span></p><p class=MsoNormal><span style='font-size:16.0pt'>spike <o:p></o:p></span></p><p class=MsoNormal><span style='font-size:16.0pt'><o:p> </o:p></span></p></div></body></html>