> 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

> 2^{k-1}(2^{k} - 1), for some *k* > 1, where 2^{k} - 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

> 2^{k} - 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