a^n, b^n, and c^n are examples of "x to the power of n". 

Importantly, it is given that all x^n are accumulations of shells according to the specific formula for that power of n. 

 means "the summation of" the 1st through xth numbers applied to the formula  S_n .

 S_n represents the shell formula that is correct for the power of n.  See shell formulas and shells envisioned.

^{x^n =}

 So   +=

Jump to the most interesting diagram where two add to a third.

If a < b < c the definition of accumulation from the 1st to the c^th shells:

Figure 1.  Graph of possible internal additions of a^n + b^n, a<b<c.

Jump to the most interesting diagram where two add to a third.

In the way that shells accumulate, the amount of a^n is in b^n, assuming b^n is the larger. 

x represents "extra" shells that boost a^n's shells up to b^n.

^{where  =  in value and + =   b^n   in value}

^{because + =}          

Jump to the most interesting diagram where two add to a third.

Table 1.  Amount values and numbers of shells of various parts of a^n + b^n.

Figure 2.  Two dimensional graph of the relations of a, b, c, x and u.  The spacing is not absolutely accurate.  Values of a, b, a^n and b^n change positioning, but not logic.

If u does not equal a^n, a^n + b^n will not equal c^n.  The reasoning is basic:  a^n and b^n share the same first shells since they are within the same power.  b^n has a few more shells  than a^n--an additional "x" value.  a^n  plus x equals b^n.  So if b^n exists first, a^n which adds to it must be equivalent to u--the last shells which add to b^n to make c^n.  If a^n exists first, then b^n should still be thought of as an a^n amount which requires an x-shells-amount to become b^{^n}.  (c^n  subtracting (2*a^n) equals x.)  The x shells may always be thought of as the (a+1)^th through b^th shells.  So u shells must equal a^n and may always be logically definable as the last shells of the set of shells that are c^n's shells.

An argument that a number of shells may be followed by "a^n-worth" number of shells, followed by an "x value" number of shells, is fine and conceivable, but moot in naming last u shells.  By definition, b^n and a^n must be permutable to "a^n plus x plus a^n value" due to the definition of b^n.

[Also be careful if considering an x that occurs after two values of a^n because thinking about 2a number of shells can be easily misunderstood in the shell/value-of-shells relationship.  2*a^n will not tie up consistently with a (2*a)^th accumulated shell value except at the power level of 1.  It is necessary to watch the roles of shells in order of occurrence which will then maintain levels "at par" with values of accumulated shells.  Correctly employing the logic of x^th-ness means seeing shells as shells and accumulations as accumulations but also relating them to each other.]

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Last modified: 12/16/05