This is about positive integers, 0 < a < b < c (and other variables that are integers)

Like everyone, I have sensed that Fermat's Last Theorem is true.  It was in a rather geometrical way--when I saw the picture of the shells or nexus numbers that add up to form cubes:

The proof was somehow already there.  Shell/nexus numbers, that definitively "grow" an xⁿ value, are a grasp-able concept that does lead to a proof-by-contradiction for Fermat's famous statement, which is roughly:

Figure 1.  For aⁿ + bⁿ = cⁿ, n must be 1 or 2 to have integer solutions;

                 In fact, n > 2 cannot have integer solutions for a, b, and c.

Formulas for shells for each power level are visibly related to Pascal's Triangle as is established in the binomial theorem.  The Powers' Pattern also shows a basis of formulas by very interesting depictions of accumulations.

Figure 2.  The first several shell/nexus number formulas.

1       = "Shell" difference formula S₁ for x^1

2k – 1       = "Shell" difference formula S₂ for x^2

3k^2 – 3k + 1       = "Shell" difference formula S₃ for x^3

4k^3 – 6k^2 + 4k – 1       = "Shell" difference formula S₄ for x^4

5k^4 – 10k^3 + 10k^2 – 5k + 1       = "Shell" difference formula S₅ for x^5

6k^5 – 15k^4 + 20k^3 – 15k^2 + 6k – 1       = "Shell" difference formula S₆ for x^6

etc.

Importantly, within the very concept of nexus number rests the fact that "shells" add up to create any xⁿ value including the aⁿ, bⁿ and cⁿ within Fermat's Last Theorem.  If cⁿ serves as the total value of two, summed, power-of-n values, then the larger of the two values only further completes the lesser value's initial run of shells.  Importantly, due to the commutative law of addition, either of the two terms may occur firstly or secondly.

So, assuming integers,  0 < a < b < c  and n > 0

and assuming  aⁿ + bⁿ = cⁿ

...the commutative law of addition as well as the shell/nexus number nature of any integer raised to a power are the basis of Figure 3.

Figure 3.  Diagrammed shells accumulate to a cⁿ total.  Either aⁿ or bⁿ may occur firstly in shell/nexus number sets, due to the commutative law of addition.  S-sub-n represents the specific formula that generates shell/nexus number values for the power of n.  The large sigma symbols indicate the procedure:  apply integer number or variables, from the value located at the symbol's bottom through the variable located at the top, to the S-sub-n formula; sum-total values to establish a bracket's value.  Shells are 'counted' by variables located on the top brackets.  a + x + u = c.  a + x = b.  The equation at the bottom depicts known and/or assumed values of bracketed shell sets.  "x-shells-value" is equivalent to [bⁿ value less aⁿ value].  Carets "^" are placed within terms to clearly reinforce that the variable of that term is raised by the power of n.

The model that is Figure 4 diagrams all possible accumulations of significant shell-sets comprising a cⁿ value pictured in Figure 3. 

Figure 4.  List of possible accumulations of a, x and u amounts of shells.  Results are listed every time a choice must occur.

Beginning with no choice:                               {0,0,0}:

First choice--possible results:              {a,0,0};{0,x,0};{0,0,u}:

Second choice--possible results:          {a,x,0};{a,0,u};{0,x,u}:

Third result:                                                   {a,x,u}.

Six different paths may be taken to accumulate a + x + u = c number of shells that equal cⁿ value.¹

Most of the paths of Figure 4 depict (aⁿ + bⁿ) or (bⁿ + aⁿ) that equal cⁿ. 

2(aⁿ) + (bⁿ - aⁿ) = cⁿ  OR  2(aⁿ-value) + (x-shells'-value) = cⁿ is also depicted.

Three sets of shells--a, x and u sets--may be graphed as lengths each extending in one certain direction.  Each direction should be traveled only once to culminate in c total shells.  This is a special three-variable vector.  Figure 5 is special because it depicts non-Pythagorean-Triangle-logic where x and y number of shell lengths are legs of triangles that SUM to an (apparent) hypotenuse that totals simply, x + y, as a resultant length.  This is because shells (only) accumulate in number.  The model of figure 5 forces a linear accumulation onto an apparently non-linear depiction because it honors three directions that represent subsets necessary within addition of two values of a power.  The model graphs valuation effects on shell-set values according to their placement within particular, implicated, orders-of-accumulation.¹

Figure 5.  Path choices in three directions, each direction 'taken' once, in order to accumulate any order of a + x + u, or, c total, number of shells.

Figure 6.  Diagrammed number of shells based upon the logic in Figure 5, with shell-set values also indicated, accumulate to a c-shell total which is also a cⁿ total value.  All six significantly possible orders of occurrence are shown--all of which must be true statements if the commutative law of addition is honored.

The sketches in figure 6 depict shells in any possible order of occurrence of a, x and u numbers of shells regarding two terms that add up.  Logically, shells must be considered in their first to last order.  Shells are ordinal objects.  So lengths within the diagram have two simultaneously applicable categories of name:  1) the number of shells which is an amount known from first to last; and/or, 2) a value of shells that a diagrammed length represents--specific portions of specific shell series for the specified power.   Order of occurrence affects values of shells.  For both categories, the summing of legs determine "hypotenuse" "lengths/values".

By the time that the vector that is logical travel from {0,0,0} to {a,x,u} is computed, the catecorner length that is central within the 3D diagram depicts a + x + u = c number of shells or sets of shell values that sum up to a cⁿ value, diagramming all six significantly possible orders and computations of shell sets.

!!!But triangles in this model also correctly solve according to the right-angle rules of the Pythagorean Theorem--if shell-value lengths are treated as if they are already-squared roots.  So square root values of shell values sets for any power are demonstrating the Pythagorean Theorem!!! 

Thus, figures 5 and 6 simultaneously and correctly depict both non-Pythagorean and Pythagorean logic where additive depictions, in translation, should also adhere to the Pythagorean rules for right angled triangles where squared sides add up to the hypotenuse.

The Pythagorean Theorem by one of its proofs

The Pythagorean Theorem is provably true.  Simon Singh gives one proof in his Appendix 1, page 287, of Fermat's Enigma:

Figure 7.  A proof of the Pythagorean Theorem.

Figure 8.  Application of the Pythagorean Theorem for right angles to values of a-, x- and u- shells sets in Figure 6 (which are based on an assumption that:  aⁿ + bⁿ = cⁿ is possible).  On the left, the diagram from the Pythagorean Theorem proof of Figure 7 is given.  On the right is the application of the the logic of the shell-sets of Figures 5 and 6--which applies to all power levels--as participants subject to the Pythagorean Theorem.

If aⁿ + bⁿ = cⁿ is assumed possible, then shell-set values as diagrammed in Figure 6 will have to obey the Pythagorean Theorem by the square roots of three complete shell-set values as well as the square root of their total. So Sqrt aⁿ, Sqrt bⁿ, Sqrt cⁿ  must evaluate as a Pythagorean Triple.  The following equations which were established in the proof for the Pythagorean Theorem are reinterpreted more comprehensively:

Figure 9a. 

(a + b)²  = c²  + 2ab

restates as:

(Sqrt aⁿ + Sqrt bⁿ)²  = (Sqrt cⁿ)²  + 2(Sqrt aⁿ)(Sqrt bⁿ)

Figure 9b.

a² + b² = c²

restates as:

(Sqrt aⁿ)² + (Sqrt bⁿ)² = (Sqrt cⁿ)²

Notice:  If n = 1, then simply:  a + b = c.  If n = 2, then a, b, and c, are Pythagorean Triples and give the theorem as it is commonly known. 

Figure 9c.

Also apparent in Figure 6:

2a² + (b² - a²) = c²  (a value and u value [which are the same] plus x value = total)

restates as:

2(Sqrt aⁿ)² + (Sqrt (bⁿ - aⁿ))² = (Sqrt cⁿ)²

So what happens if n is larger than 2?

Since the square roots of each of aⁿ and bⁿ are "lengths" that will add up, after squared, to the "length" that is the square-root-of-cⁿ-when-squared (according to the Pythagorean Theorem), these lengths in terms of an p, q, and r that are p² and q² that add up to r² are, [the square root of aⁿ]= p, [the square root of bⁿ]= q, and [the square root of cⁿ] = r.  But if the three values are Pythagorean Triples then:

Figure 10.

Also note:  square roots of each of a, b, and c do function when n = 1, because the square roots square to the values of a, b, and c, and the accumulation is simply linear.

For (Sqrt aⁿ)² + (Sqrt bⁿ)² = (Sqrt cⁿ)²

that must describe aⁿ + bⁿ = cⁿ, if it is true,

there is no possible integer solution when n > 2.

The Pythagorean Theorem limits the Pythagorean Theorem

due to the commutative law of addition and the fact that shells/nexus numbers always sum to a cⁿ value.  Shells establish functioning subsets that do not allow cⁿ to be divisible into "two powers of the same denomination" when the power level is greater than 2.

Cubum autem in duos cubos, aut quadrato-quadratum in duos quadrato-quadratos, et generaliter nullam in infinitum ultra quadratum potestatem in duos eiusdem nominis fas est dividere:  cuius rei demonstrationem mirabilem sane detexi.  Hanc marginis exiguitas non caperet.   

To divide a cube into two other cubes, a fourth power or in general any power whatever into two powers of the same denomination above the second is impossible, and I have discovered a truly marvelous demonstration of this proposition that this margin is too narrow to contain.

--Pierre de Fermat

Regards, Cecilia 

October 1, 2005

¹  Incidentally, this type of model holds for any number of examined subsets when order matters to the counted amounts within sets, and when choices are cumulative to one single, largest set.  Values of Pascal's Triangle row z [do not count the top "1" of the triangle] also called C(z,0 to z-1) always enumerate the number of possible results at each of the 0th through (z-1)th choice.  There are always z! possible paths going from no sets to all sets.  There are always 2^z possible subsets of z sets within the total, if you include 'no shells' as a subset.  Finally, path diagrams look like sketches of z-dimensions.]

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