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10860Re: [GTh] Triangular and Perfect Numbers

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  • Mike Grondin
    Mar 22, 2014
      > Actually the ancients from the time of Euclid knew in principle how
      > find perfect numbers. A number will be a
      style="TEXT-ALIGN: justify; TEXT-TRANSFORM: none; BACKGROUND-COLOR: rgb(255,255,255); TEXT-INDENT: 0px; FONT: medium 'Times New Roman'; WHITE-SPACE: normal; FLOAT: none; LETTER-SPACING: normal; COLOR: rgb(0,0,0); WORD-SPACING: 0px">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.) See for example
      What the ancients lacked were the means to determine whether or not
      >   2k - 1 is prime for large k.
      Thanks for this correction, Andrew. For those who don't take the trouble to
      read the excellent article to which you link, the k-value of the 5th perfect number
      is 13, so it's 2**12 (2**13 - 1) = 4096 * 8191 = 33,550,336 (which gives a further
      indication of how rare perfect numbers are). It's not that k is too large, but that
      they apparently lacked the means to determine something else. But what was that
      "something else"? To determine that 8191 is prime requires only that one know
      which numbers below 91 are prime (because the square root of 8191 is somewhat
      less than 91). Seems that they would have known that. Maybe it was the difficulty
      of determining all the factors of 33,550,336 or maybe they had checked k=11 and
      determined that that didn't yield a perfect number, so they gave up?
      In any case, there's apparently no evidence of the knowledge that 8128 was a perfect
      number until Nicomachus of Gerasa around 100 CE, so at the time GJn was written
      (or at least the prologue and chapter 21), 496 may have been the largest widely known
      perfect number. Be that as it may, however, it must have been considered a fortuitous
      (if not divinely ordained) coincidence by the final composers of the Fourth Gospel that 
      496 was also the isopsephic value of MONOGENHS ('only-child' or 'only-begotten'.)
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