Consider the equation c = ab Again, if a and b are unique integers, the product must be equal to some integer d. But this means that c=d, another tautology. If a and b are not unique integers, then c = k(ab) for some positive integer k. But then c = (ka)b=a(kb) so that c = a'b or c=ab', where the integers have been scaled to a specific k, so the variables are not independent, and any expression involving k will not be general.

Therefore, for generality, (a,b) must be characterized variables over the positive integers, with the scaling ratio true for all integers, so that for t and t' positive integers, (at,bt') are independent. This is a metric comparison, in which a and b are scaled, such that the scaling is true for all possible combinations. If we select at as the foundation metric, to be compared with bt', with

as the initial standard for comparison of the metrics, then the relation:

fpr at, bt' independent variables, so the scalar operation is true for all a,b,t and t'. The only way to ensure that that the equality applies for all a and b is to assume that at and bt' are independent.

Solving this gives the equation

, where

Under what conditions can all the parameters be integers?

For c

to be an integer, must be an integer, where .

, so that .

, so that .

Then if is an integer, either or so that the equation means that or .

If then since no initial metric exists for comparison. If then and the metrics are the same there is no multiplicative scaling, so that the multiplicative product does not exist. (there is no multiplicative relation between a and b))

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