it takes a long, long time to digest the following argument:
i believe that (O)dd (P)erfect (N)umbers do NOT exist in the same
manner as (E)ven (P)erfect (N)umbers do!
if we allow the generic formula of (k^(m-1))*(k^m - (k-1)) to de-
fine the situation, then we will see that it encompasses both the
proven EPN(n) == (2^(n-1))*(2^n -1) and a closely-related possible
OPN(n) == (3^(n-1))*(3^n -2).
however, we can't be satisfied this approach, since these two in-
stances of a preferred formula don't have comparable 'pole' posi-
tions when 'n' is set equal to zero.
if we let n= 0, then the 'pole' for EPN(0) == (1/2) *(0) remains
unshifted from the origin, and the 'pole' of OPN(0) == (1/3) *(-1)
is shifted by a (-1). comparing the results with respect to per-
fection between these two instances is already suspect from the
onset.
after a closer examination, though, the formula (3^(n-1))*(3^(n+1)
-2) turns out to be a more appropriate function but with a little
twist involved; if we let n= 0, the 'pole' of OPN(0) == (1/3) *(1)
is shifted by a (+1). we'll use this fact to later offset our clo-
sest OPN formula by a (-1) to bring the two 'poles' into agreement.
now, a 'proper-incipient' EPN is generated when the exponent 'n' is
less than the value inside the portion of the second parentheses of
its generic formula; n= 1 implies that EPN(1) == (1)*(1), but it's
improper since n= 1 is not less than the second (1); n= 2 implies
that EPN(2) == (2)*(3) and since n= 2< (3), we quickly discover that
6 = 1 *2 *3 = 1 +2 +3 = 6 (is an unshifted 'proper-incipient' char-
acteristic that should be shared by the preferred OPN function).
if n= 1, then the preferred OPN formula gives an OPN(1) == (1)*(7)
and 1< (7). so, the OPN(1) would share the proper 'incipient' qua-
lity of its cousin EPN, but not without an adjustment to its value
with an offset of (-1) to combat the earlier mentioned (+1) shift;
allowing for this little twist on the OPN's existance, the OPN's be-
come not only odd, but a bit quirky!
if we subtract '1' from the summand and allow the use of an 'impro-
per' divisor, then we have OPN(1) == (1)*(7) = (-1) + ((1) +7) = 7
(offsetting it), and the incipient OPN presents itself; but it was
not created using the expected 'EPN' axiom. it's the only OPN attach
ed to k= 3 from the preferred OPN formula, but not the only OPN...
if k= 5 and n= 1, then 21 isn't the next OPN, but...
finally, if all OPN's are == (2^(n-1))*(2^(n+1) -1), or (3^(n-1))*(3
^(n+1) -2), or (5^(n-1)) *(5^(n+1) -4), or (7^(n-1))*(7^(n+1) -6), or
(11^(n-1))*(11^(n+1) -10), or (13^(n-1))* (13^(n+1) -12), etc., then
we can see that the sequence of OPN's intermittently emerge as 3, 7,
(not 21), 43, (not 111), 157, etc.
the OPN's are not governed by the same rules as their cousin EPN's &
are not the expected 'gigantic' numbers that contain a multitude of
factors, but actually a set of individual (must be) prime numbers
that are generated by a continually changing formula and by modify-
ing the calculations of both the product and summations due to a
shift from their respective 'pole' positions.
thus, OPN's could barely exist but not in the usual EPN sense... they
are quirky! at best, the term OPN is a fictional misnomer, because
all possible formulas for an OPN lack the offset that is necessary to
coincide with the zero 'pole' of an already Euler-proven EPN formula
which also share the 'proper-incipient' characteristic.
the definition has to be changed... re-read it!
Bill Bouris |
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