**It is impossible to prove Fermat's theorem without two independent number lines representing a and b, because there is no consistent distinction between the partitions within the same set**, e.g. for a = 24,

, etc., etc., etc.....

All the coefficients are simply ways of counting all the elements of the set represented by 24 - which is the single element (positive integer) a = 24 represented in the equation for Fermat's Theorem:

for a,b,c and n positive integers, n > 2

(If there is only one number line, there is no second element "b")... it is not that b = 0, but that b does not exist in the equation, it is only a meaningless symbol and all one has is

.... with no relationship specified except equality.

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