>ratio of primes is 1/ln n, so I would
>assume that as you approached (inf,inf,inf) you would
>find coordinates (x1,y1,z1) and (x2,y2,z2) such that
>ln x1y1z1 - ln x2y2z2 is arbitrarily close. Any takers
>on a proof or disproof?
ln x1y1z1 - ln x2y2z2 is minimum when x1y1z1-x2y2z2=2.
Assuming this happens infinitely often, then ln x1y1z1 - ln x2y2z2 tends to
From: Leadhyena Inrandomtan [mailto:leadhyena_inrandomtan@...
Sent: 31 January 2002 20:45
Subject: [PrimeNumbers] About that coordinate system...
I apologise if I break any rules of netiquette,
as this is my first send on this post. I am but an
independent mathematician out of Dallas, TX and have
an insatiatable appetite for mathematics and logic. So
here are my two cents on the prime coordinates issue:
There is a way to make the addition of
coordinates work, but you step outside the realm of
integers by doing so. Taking upon the idea presented
by Sergio Ribeiro, you could take the main X, Y, and Z
poles and use the ln of all of the coordinates. Then
by adding X+Y+Z them you get unique coordinates for
all normal (X,Y,Z). Reason is due to Unique
Factorization Theorem: just like multiplied
coordinates in Sergio's system have a unique
factorization, the addition of the three ln
coordinates also equals ln XYZ and this number is
always unique by construction.
I'm not sure if it can be scaled to integers,
using a system such as floor(C*(ln X + ln Y + ln Z)),
finding a C that would give each coordinate a unique
integer value. The reason is due to the distribution
of primes: ratio of primes is 1/ln n, so I would
assume that as you approached (inf,inf,inf) you would
find coordinates (x1,y1,z1) and (x2,y2,z2) such that
ln x1y1z1 - ln x2y2z2 is arbitrarily close. Any takers
on a proof or disproof?
I couldn't forsee the use of such a system,
however. Cantor already proved that Z^3 and Z are the
same cardinality and this would also prove the same,
albeit more romantically (not necessarily more
elegant, but more aesthetically pleasing ;). You
already have a real number for every R^3 coordinate,
in a somewhat more dense fashion. I could see the
interesting possibility of adding two ln XYZ
coordinates together, because the pair of coordinates
added together could be deconstructed from the sum,
using the same reasoning above. (three coordinates
however IS impossible by the presented method)
As far as the philosophical debate, I say
whatever floats your boat. What matters except for the
mathematics? Why step outside of the mathematics? It's
only symbols; It's inherent in nature; It's man-made;
it's created by God; Man is God whatever. What matters
is the mathematics. To put in my two cents though,
this is how I see it: Logic is god, and reality is
just the symbols of god flying around to the winds of
logic. No personification. Mathematics is simply the
practice of putting symbols on paper and letting them
dance on the winds of logic. But does it matter to the
mathematics? As long as math doesn't try to supercede
logic, we're on solid ground.
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