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-   -   Affine transform between infinite dimensional quantized space and 1d scalar space (http://mymathforum.com/number-theory/336578-affine-transform-between-infinite-dimensional-quantized-space-1d-scalar-space.html)

BenFRayfield October 15th, 2016 11:46 AM

Affine transform between infinite dimensional quantized space and 1d scalar space
 
e^sum(log(integerA)+log(integerB)...) = integerA*integerB*...

log(integerA)+log(integerB)... = log(integerA*integerB*...)

Every unique bag (set that allows duplicates) of primes multiplies to (and factors from) a unique positive integer.

Like in Godel Numbering, if dimensions are numbered by nthPrime(dimensionIndex), and position in dimensionIndex is how many of that prime are factors, that is an infinite dimensional quantized space (a map of integer to integer).

An infinite dimensional quantized space is affine transformed to 1 dimension (and back again) by multiplying position in each dimension by log(nthPrime(dimensionIndex)).

Example: log(2)*integerX + log(3)*integerY + log(5)*integerZ = log(2^integerX * 3^integerY * 5^integerZ)

These sum to unique scalars with exponentially increasing density, the log of each positive integer exactly once.

This appears to somehow relate Factoring to Subset Sum except not of integers.


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