The Related Numbers of Positive-Integer-Power Series

Presenting a "Cube" of the Figurate Numbers of Pascal's Triangle and Euler Triangle Numbers

Then examining the road to an Elementary Proof of Fermat's Last Theorem

Here is a concise FLT proof and click here for other FLT musings

By observing the figurate content of sieved accumulations from the integer series to powers series, the reality of a cube of series' relationships comes to light.  The relative meaning of powers is explained here.  Two (logically reciprocating) formulas in ten figures follow that describe relationships of series within the Euler/Pascal cube.  See another essay for more on sources of the formulas.  Series that are the powers (or, xⁿ values,) in the Euler/Pascal cube are fully described and defined by Worpitzky's identity of 1883.  (Other series within the depicted cube of series may be considered a logical expansion of Worpitzky's identity.)

Figure 1.  Formula "SeriesAtLevelR" (as produced with the Mathematica program, from Wolfram Research, Inc., using an 'add-in'):

<<DiscreteMath`Combinatorica`

Figure 2.  Layout of variables for a diagram of "SeriesAtLevelR":

Figure 3a.  Some initial solutions for the formula "SeriesAtLevelR" using the layout of figure 2:

Figure 3b.  The pattern extends forward (also by implication of figure 4's formula.)

Figure 4.  Formula "MagicNKZ" for a (k+1)^th value of power level n at z level of accumulation within the Euler/Pascal Cube (as produced with the Mathematica program, from Wolfram Research, Inc.):

Figure 5.  Layout of variables for a diagram of "MagicNKZ":

Figure 6.  Solutions for the formula "MagicNKZ" using figure 5's layout (notice Euler's Triangle on the front and Pascal's Triangle on the top):

Figure 7.  The following formulas illustrate the influence of the Euler Triangle across power series.  Each formula applies to a horizontal layer of the cube as pictured in figure 3 [and 6].  There is one specific power level addressed per layer.  (Produced using the Mathematica program, from Wolfram Research, Inc..)

If r = -1, x solves for shells/nexus numbers to powers.  If r = 0, x solves for values that are powers and this is Worpitzky's identity of 1883.  If r = 1, the solutions are summations of the 1st through xth of a power.  (If r = 2, the solutions are 1st through xth summations of summations of powers, and the pattern continues for r.) 

The coefficients to the terms in figure 7 at level n are the values from a Euler Triangle row each multiplying n neighboring values of a figurate series of the Pascal Triangle.  The "Euler weighting procedure" applies across the figurate numbers series for a single power's shells of shells series through its summations of summations series.  Since figurate numbers change in a predictable manner, and since the single Euler row "treatment" defines shells of a power as well as powers; and since the commutative laws of addition (see figure 9) are true in the addition of powers, Fermat's observation, that aⁿ + bⁿ cannot equal cⁿ when n > 2, is very simply true.

Figure 8.  A diagram of examples of the "Euler weighting procedure".  Click for another type of diagram.

REGARDING FERMAT'S LAST THEOREM  click for a different FLT summary

Figure 9.  When two values of a power level add successfully to make a third value within a power level, they will add up to a final complete set of shell/nexus numbers that accumulate to the value that is c to the power of n.  A "shell" value is a value that is the difference between the (x+1)th and the xth value of a power.  The 1st through xth shells of a power n accumulate to a value that is xⁿ.  Because of the commutative law of addition, some latter shell(s) of an ultimately cⁿ shell-series will have to be equivalent in value to both possibilities of:  the lesser of the two terms' value (depicted below as aⁿ), as well as, bⁿ, which has some additional middle shell(s)' value added to the aⁿ value.  (The commutative law states either of two proposed terms must add either firstly or secondly, i.e.,  aⁿ + bⁿ or bⁿ + aⁿ should equal cⁿ.)  In the below figure, S-sub-n represents an equation (easily determinable from Pascal's Triangle) defining any xth shell for the power of n.  cⁿ represents the summation of the 1st through cth shells.

Figure 10.  The figurate numbers of Pascal's Triangle are important in the definitions of both powers and shells of powers according to the Euler/Pascal relationships.  The kth of any Pascal Triangle figurate sequence equals the total sum of the 1st through kth of the previous sequence.  All series, except the first figurate sequence, which is 1's, and the second, which is integers, increase by an increasing rate within its values.  Acceleration in the increase of value within series occurs by a rate between two series that is the first sequence's equation multiplied by a multiplier that is:  (k + z - 2) / (z  - 1) where z is the new zth level of figurate series involved.  The first figurate sequence, or z = 1, is given as entirely 1's. 

n contiguous values of the (n + 1)th-figurate level treated by the nth-Euler-Triangle-row in the Euler/Pascal/Worpitzky (weighting) relationship equals an xⁿ value, AND; xⁿ equals the sum of the 1st through the xth of the shells of the power of n.

[n contiguous values of the nth-figurate level, treated by the (same-as-in-the-previous-paragraph) nth-Euler-Triangle-row, according to the Euler/Pascal/Worpitzky (weighting) relationship, equals a shell value.]

Due to an increase of increase of values in figurate series, no single shell/nexus number value, when n > 2, will be equivalent to a power value since no three or more values in a row, upon which a shell definition will depend, will be the same as the three or more values in a row upon which the power definition will depend--both would be treated exactly the same by Euler Triangle row n.  Higher figurate series are accumulations of lower series.  That limits the amount of matching values betwixt neighboring series to rare and assuredly non-neighboring cases.  So, as pointed out in Figure 9, a latter shell might begin to satisfy an effective equation if it would equal the value of a power.  Obviously though, this cannot happen for any single shell of the powers of three and higher.

(For the power of two, odd numbers 1,3,5,7,9... are shells so, of course, every odd xⁿ value will have a matching value within the shell series.)

It should be mentioned that finding equivalent values between the shells (or a contiguous set of shells not including the first) and a value of the power is a necessary but not sufficient requirement to establish that aⁿ + bⁿ = cⁿ.  For the case of Fermat's Last Theorem, it is only important to establish there is a failure to meet the absolutely basic requirement that some latter shells be equal to an xⁿ value. 

As stated above, obviously and evidently, no single shell can serve as equivalent in value to a power's value for the power of 3 (or greater).  Might a latter set of shells correctly weight to equal an xⁿ value?

Try to imagine n amount of the (n+1)th-figurate numbers, each having shorted values because of non-included firstly occurring values of the nth-figurate numbers.  (Missing first shells means less value per (n+1)th-figurate number used to define cⁿ due to figurate numbers defining the "missing" firstly added value).  For a missing shells, the cth of the (n+1)th-figurate series values is shorted the 1st through ath of nth-figurate series values; the (c-1)th of the (n+1)th-figurate series values is shorted the 1st through (a-1)th of nth-figurate series values; the (c-2)th of the (n+1)th-figurate series values is shorted the 1st through (a-2)th of nth-figurate series values; etc., continuing in the same manner for all of the (n+1)th-figurate values within the definition.  So n amount of short-shrifted (n+1)th-figurate values would then be weighted by multipliers that are Euler Triangle's row n and summed.  The result would accurately define the total value of a latter shell-series set.  However, the value would be a value neither here nor there in terms of the values of a power series.

The reason:  n amount of contiguous values of the (n+1)th figurate level numbers, with a latter-shells' "missing-some-1st-shells-situation" that short-shrifts each of the n values, will never be any one of the known and correct proportions within n contiguous figurate values that DOES "weight" correctly by Euler Triangle's row n and which leads to each and every xⁿ value.  The values that would be proportionally correct for every power-value are "known".  The proportions are those that precisely and constantly increase by increasing amounts (non-linearly) at the rate of the (n+1)th figurate level as seen in its equation.  All sets of values presenting observably correct proportions between and among its values must be expressible as (contiguous) integer solutions of one of the figurate equations of Pascal's Triangle (depicted in Figure 10). 

When subtracting the early amounts out of a sequence of increasing values in non-linear series, the acceleration of the acceleration, or, the equation that solves for the values, is skewed for the new, shorter, series.  This means that the proportion of each term of a set compared to other terms--not to mention the proportions of the terms compared to their sum total--becomes "guaranteed to be not representative of any of the proportions of terms as found across the (n+1)th-figurate series that assuredly define power values by the Euler-row n weighting-level".  Short-shrifted, proportionally incorrect terms never match any proven-to-be-correct distribution that must be describable as:  the distributions as they appear in any n, contiguous amount of values of the (n + 1)th figurate series.  

The exception:  sometimes some contiguous, short-shrifted, 3rd-figurate series values (that define latter shells for the power of two) and a regular neighboring pair of 3rd-figurate series values (that are in the definition of a power-of-2 value) actually MATCH in value!  This is due to the fact that the sequence that is the power of 2 accelerates at a linear rate in its values (which means shells have a linear formula.)  (Strategically, the Euler-weighting of the nth-, 2nd-, [integer] figurate series is translatable into a summation of a linear series of values.  Click to see more on this.)  The power-of-2 exception also succeeds due to the fact that the Euler Triangle row 2's weights are only (innocuous) coefficients (weighting by 1 and 1) which allow the distribution of value within or between terms to be any distribution.  Some  a² + b² do equal c². 

Fundamentally, just as everyone senses, there is no hope for two of higher powers to add correctly to cⁿ due to the accumulative nature of powers visible in the non-linear growth of the figurate sequences that are so important to the definition of power values.  The Euler-Triangle-row treatment of differently non-linear neighboring figurate series within definitions of power values NIXES the possibility of any set of contiguous shells of a power summing to equal a power value--unless the set includes all of the 1st through xth shells.  (Euler/Pascal definitions of shells and powers are recurring in nature and support this). 

So, because of the denial of the commutative properties of addition which must apply (to two sets of shells of powers as they accumulate en masse to a cⁿ power value--see Figure 9), two values within a power will never add to a third of that power if the power is 3 or greater.

Relationships of numbers within the Euler/Pascal cube will very likely obviate the nature of other conjectures in the same way that it illumes the basis of Fermat's 17th century "unproven" statement:

Cubum autem in duos cubos, aut quadratoquadratum in duos quadratoquadratos, et generaliter nullam in infinitum ultra quadratum potestatem in duos ejusdem moninis fas est dividere:  cujus 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 assuredly found an admirable proof of this, but the margin is too narrow to contain it.

--Pierre de Fermat

Summary of FLT

click for another FLT summary

A definition of a value of the powers may be based on the figurate numbers of Pascal's' Triangle according to Worpitzky's identity of 1883.

Two pertinent facts about the figurate numbers:

1) The 1^st through x^th of the n^th–figurate series, summed, equals the x^th of the (n+1)^th–figurate series.

2) The x^th value of any figurate series after the second will be describe-able by a non-linear equation, and be different for each figurate level.

If the nature of the values of the (n+1)^th figurate series, upon which a power's value is based, is split between two terms to be added—say, aⁿ + bⁿ—the secondly occurring term will depend proportionally upon the (a+1)^th through c^th of the n^th-figurate series.  In comparison, the first term regards the 1^st through a^th portion of the series precisely as described by the Worpitzky definition for aⁿ.

However, the proportions of two terms mean:  the secondly added term cannot be a power value because proportions that are wholly (n+1)^th–figurate values define each and every xⁿ value (as visible in the Euler/Pascal cube of relations).  No partial, secondly added, portion-of-(n+1)th-figurate-values will be amounts that are (n+1)th level figurate numbers necessary to the definition of a "bⁿ" value.  Only entire, non-partial sets of 1^st through b^th values of the n^th–figurate-level will define a value of the (n+1)^th level--for levels of figurate series of Pascal's Triangle that increasingly accumulate at non-linear rates of increase within the values of the series:  which means when n=3 or greater.

For levels and powers of 1 and 2, some (a+1)^th through c^th of n^th–figurate values may equal the 1^st through x^th of n^th–figurate values!  (Furthermore, Euler-weighting/distributions for powers do not really function as "definition-constraining distributions" until the power of three.) 

If both of two proposed terms successfully define power values (by the Worpitzky/Euler/Pascal relationship as well as by the summations of shells of a power definition), and both terms may equal certain latter shells of the 1st-through-cth-shells-set addressing the situations that are the addition of terms either firstly or secondly, then, and only then, aⁿ + bⁿ may equal cⁿ.  This never happens for the powers of 3 and higher.

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