## 10860Re: [GTh] Triangular and Perfect Numbers

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• Mar 22, 2014
> Actually the ancients from the time of Euclid knew in principle how
to
> 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'.)

Mike
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