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10857Re: [GTh] Triangular and Perfect Numbers

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  • sarban
    Mar 22, 2014
       
      ----- Original Message -----
      Sent: Friday, March 21, 2014 6:39 PM
      Subject: [GTh] Triangular and Perfect Numbers

       

      By now, you all will have received notice that I've uploaded a new file
      called 'Numbers.xls' to our Files section. This file shows the first four
      perfect numbers (the only ones known in antiquity) and the triangular
      numbers of 3 thru 36, with notation as to the association of some of
      them in canonical literature and Coptic Thomas.
       
      Bauckham remarks that perfect and triangular numbers are rare. This
      is misleading. Perfect numbers (i.e., numbers equal to the sum of their
      divisors or factors, including 1) are much more rare than triangular
      numbers. Furthermore, the ancients had no idea how to find perfect
      numbers, but knew quite well how to find triangular numbers.
       
       
      Actually the ancients from the time of Euclid knew in principle how to find perfect numbers.
      A number will be a perfect number if of the form 2k-1(2k - 1), for some k > 1, where 2k - 1 is prime.
      (They could not prove that all perfect numbers are of this form but this has still not been rigorously established.) 
      What the ancients lacked were the means to determine whether or not   2k - 1 is prime for large k.
       
      Andrew Criddle
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