Infinite dimensional space to represent primes
Date: Around 1995
wondering if there is any literature on this space that I am thinking about.
Suppose that for each prime number I create an axis. The value on an axis
is a non-negative integer. A point in the space represents a number
determined by the coordinates of the point. For example the number 1 would
be 0, 0, . . . or 2^0 * 3^0 * 5^0 . . .)
And so 6 would be represented as
(1, 1, 0, 0, . . . or 2^1 * 3^1 * 5^0 * 7^0 . . .)
Is this idea of any use other than something fun to think about?
What you are really describing is an infinite dimensional hyper-cubic unit
lattice. The lattice points correspond to the natural numbers. The
projection of a lattice point onto an axis tells you the number of times the
prime represented by that axis divides the number represented by the lattice
point. Primes correspond to lattice points that have a single 1 in their
coordinates. The weird thing is (if this is all correct) is that the
cardinality of the lattice points has to be the same as the natural numbers,
i.e., Aleph nought!
Well, John's comment on cardinality is true only if you allow only finitely
many of the integers in the lattice points to be non-zero. I assume the
cardinality of the whole thing is essentially the same as the real numbers,
if you do allow infinitely many non-zero integers (but any time you have
infinitely many of non-zeroes, you no longer are representing a natural
number - the product you get is infinite!) Anyway, it is certainly a cute
idea. Note that you get all the rationals by allowing negative coefficients
(all the positive rationals, that is), and there is actually a unique
mapping of each rational number into this space (and it is 1 to 1, if we
leave out the infinite non-zero sequences). Maybe somebody more well-versed
in number theory will tell us all the results derived this way.
One question is: "What advantage can you get from this representation?" I
would guess that there is certainly no geometric advantage for visualization
be cause of the infinite number of axes. We are dealing with infinite
sequences of integers so they can be added vector style and one can form
integer scalar multiples of the "vectors." Addition should be the same as
multiplication of the integers represented and scalar multiples should
correspond to powers. We do not have a vector space because the scalars
must be integral, but maybe an R-Module?
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Update: June 2012