POLARITY-INDUCTION OMISSION STRATEGY

A SPECIAL CASE SERIES for n^2 + 1 = NP

by Brian S. McMillan

NEW POSTULATE (Brief)

Where any Odd Integer is a Common Factor of the Exponents for at least two of the terms in a Three Term Equation, then the Sum of those two terms when raised to their respective Exponents will always equal a Composite Integer, and therefore, may never be Prime. Since this is easily demonstrated by dividing the Sum of the bases for those two terms as raised to the remaining factors for their respective Exponents, that is less the Odd Common Factor, then we now have the basis for our argument.

Since the following examples may contain any combination of integers to represent the bases within the terms, and it has been known for some time that this type of argument may be applied using Odd Exponents only, and with the exception of the specific case for Fermat Numbers and theoretically any Integral base raised to the higher powers of 2, this stands as the more generalized proposition in the case of both Even and Odd Exponents. A proof for The Last Theorem of Fermat is therefore implied, and may be reduced to a single three term equation. Furthermore, a strategy for both the Goldbach and Opperman Conjectures is likewise revealed. Thank you.

The New Postulate for Even as well as Odd Exponents, completely removes the necessity for recognition of the sine or properties of the Integers which comprise the bases of the terms and will work equally well for any combination of Even or Odd bases through Subtraction as well as Addition between the terms. Any equation which may incorporate the more Modular Argument for the Fermat Little Theorem as well as the Mersenne Series may also be addressed, and therefore establishes a Proposition which is totally independent of the considerations of Primality or Co-Primality within the bases of those same terms, and should ultimately reveal the very underpinnings for the operation of those mathematical strategies. However, as with any theoretical approach, there may be instances where there is a convenient overlap in our ability to represent a proof. Happily, there is overlap-a-plenty. This will become clearer a little farther down this page while using a combined Polarity-Property Strategy. While this may ultimately be unnecessary, it will provide a basis for further proof of the arguments.

Note: Euler and Lucas are both credited with a proof for the denumeration of Fermat Numbers greater than F4, however these proofs have not convinced everyone. This is, theoretically, the only remaining impediment to a truly brief and simple proof for the (so called) Last Theorem of Fermat.

Because n^2 + 1 may only be Odd when n = Even Integer

A Special Case for the Exponent when equal to a (4k + 2) series Integer.

WHERE: Odd is Greater Than 1 for all Examples

IF: (4k + 2) = 2 * Odd

AND: (a^(2*odd) + b^(2*odd)) /(a^2 + b^2) = Integer

THEN: ((Even^odd)^2 + 1) /(Even^2 + 1) = Integer

WHERE: (Even^odd) = n

HENCE: n^2 + 1 = NP_For Some n = Even^odd

A Special Case for the Exponent when equal to a (4k + 0) series Integer, where k = Odd, a general example may be represented thus.

IF: (4k + 0) = 4 * Odd = 2 * (4k + 2)

AND: (a^(4*odd) + b^(4*odd)) /(a^4 + b^4) = Integer

THEN: ((Even^(2*odd))^2 + 1) /(Even^4 + 1) = Integer

OR: ((Even^(4k+2))^2 + 1) /((Even^2)^2 + 1) = Integer

HENCE: ((Even^(2^n*Odd))^2 + 1) /((Even^(2^n))^2 + 1) = Integer

To be continued.

New Postulate continued (brief) these are just a limited set of examples:

WHERE: x = Prime or Odd Composite, y = Even Integer, z = Any Integer

AND: a or b are greater than 1

IF: (a^(x*y*z) + b^x) /(a^(x*y*z/x) + b^(x/x)) = Integer

THEN: (a^(x*y*z) + b^(x*y)) /(a^(x*y*z/x) + b^(x*y/x)) = Integer

WHERE: x = 3 or Odd Composite containing 3 as a factor, y = Even Integer, z = Any Integer

AND: a or b are greater than 1

IF:

((a^(x*y*z) + b^x) /(a^(x*y*z/x) + b^(x/x)) - a^(x*y*z/x) * b^(x/x))^(1/2) = Integer

THEN:

((a^(x*y*z) + b^(x*y)) /(a^(x*y*z/x) + b^(x*y/x)) - a^(x*y*z/x) * b^(x*y/x))^(1/2) = Integer

The Fundamental Theorem of Integers (brief)

The Natural Integers possess fundamental, skeletal properties, which may be altered or changed by the various operations of arithmetic, i.e. Addition, Subtraction, Multiplication, Division, Roots, Exponents... Commutative, Associative, Distributive, and by default the (Primary) Identity for some of the Integers remain non-mutable within the bounds of the operations. These Integers which possess non-mutable (Primary) identities, are responsible for the dominate characteristics observed within the Even and Odd Integers. One of the many uses for this approach is embedded in the exclusion principles that these properties obey in relation to the Natural Integers, allowing or disallowing certain Integral Operations. It was Pierre de Fermat, around the historical period of 1650 A.D., which first categorized these properties. This work would not have been possible without him.

I have discovered a previously non-explored branch of Simplified Integral Algebra, or Geometry as your preference, since it is easily and economically applicable to the cardinality arguments of the Euclidian Plane, as well as interchangeable with all Gaussian Induction Formats. I believe this to be as fundamentally important as anything that has been handed down to us by the Ancient Greeks. If any of you wish to look at it, the click-on is below, and the primary table appears as a small series of equations about three paragraphs of equations from the top of the page. I've listed one of them below. I am honored to bring this to you all. Thanks Again.

Yours,

Brian S. McMillan

http://www.pumpraser.com/fermatmersenne.html

Now comes the properties for the individual terms as they might be raised to an Odd power. This particular solution may be applied as a rule of omission for the problem known as "The Last Theorem of Fermat". This rests upon the fact that while certain 4n + 1 series numbers may be root-able to the square... can 4n + 1 + 1 = 4n + 2 series numbers be root-able to the square or the cube, or to any power for that matter? For example:

WHERE: Odd + Even = Odd, Odd + Odd = Even, or Even + Even = Even

IF: (4k + 1)^even = 4n + 1

AND: (4k + 1)^odd = 4n + 1

AND: (4k + 3)^even = 4n + 1

THEN: (4k + 3)^odd = 4n + 3

OR: (4k + 0)^even = 4n

SO: (4k + 0)^odd = 4n

AND: (4k + 2)^even = 4n

THEN: (4k + 2)^odd = 4n

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We have seen above, that when we attempt to yield a (4n + 2) integer by any and all of the combinations of possible bases with their corresponding exponents, the reverse of the operation is also prohibited. So for three out of the four possible Odd integer combinations shown below, a (4n + 2) integer is the result. We may now eliminate those three combinations from our mathematical choices.

IF: (4a + 1)^x + (4b + 1)^x = 4n + 2

AND: (4a + 3)^x + (4b + 3)^x = 4n + 2

THEN: (4a + 1)^even + (4b + 3)^even = 4n + 2

SO: (4a + 1)^odd + (4b + 3)^odd = 4n

TRANSPOSITION OF BASE PROPERTIES WITHIN THE TERMS

It may be noted that the base Integers for all three terms may be altered within certain guidelines when two of the three terms are raised to an Odd Exponent. The usefulness of this approach is embedded within the number of different ways that these base Integers may be represented as well as to provide conclusive proof that The Generalized Proposition for Exponents has no dependency what-so-ever regarding the Properties for the base of the terms except with regard to the individual properties of those terms in relation to the Divisor Sum as both being raised to Even Powers, and therefore may lead to a more cogent theory of Numerical Groups. However, because this Transposition of Properties does not work for all Even Powers, this is solid testimony to the correct interpretation for The Fundamental Theorem of Integers. For Example:

For the Variable Integers (a, b) contained within the bases of the terms, we may take the Sum of (a + b) = c and subtract any set of Integers from and up to (c - 1) to yield c - d = e, so that now we have a new a and b which is still divisible by our original base Sum. For Example:

IF: (a + b) = (d + e)

THEN: ((4d + 1)^odd + (4e + 3)^odd) /((4a + 1) + (4b + 3)) = Odd

SO: (4a + 1 + 4b + 3) = (4a + 4b + 1 + 3) = (4 * (a + b) + 4)

To further explore this we may even transpose the variables themselves within the main terms. For Example:

IF: (a + b) = (e + d)

THEN: ((4e + 1)^odd + (4d + 3)^odd) /((4a + 1) + (4b + 3)) = Odd

SO: (4e + 1 + 4d + 3) = (4e + 4d + 1 + 3) = (4 * (e + d) + 4)

Finally we may now fully transpose the bases of the terms through Property Mutation. For example, if the Properties for the Base Integers is one of an Odd Sine, lets say a (+ 1 or + 3), as long as the end result equals the Key Property Sum for our original two base terms, then we may now convert these two base values to reflect the Sum of two Even Integer Types. Even though the Exponents are involved in moulding the final yield, while the total may be different, it is still divisible by our original Summand. For Example:

IF: (a + b) = (e + d)

THEN: ((4e + 2)^odd + (4d + 2)^odd) /((4a + 1) + (4b + 3)) = Even

SO: (4e + 1 + 4d + 3) = (4d + 4e + 1 + 3) = ( 4e + 2 + 4d + 2) = 4a + 4b + 4 etc.

A NUMERICAL EXAMPLE:

IF: ((4 * 31 + 1)^3 + (4 * 09 + 3)^3) = 2012444

THEN: ((4 * 31 + 1)^3 + (4 * 09 + 3)^3) /((4 * 31 + 1) + (4 * 09 + 3) = 12271 Odd Integer

For the Sum of Variables (31 + 09) = 40, we may now change any combination of those variables as long as they add up to 40, such as (20 + 20) = 40, and our original Divisor Sum will still factor our altered Yield. For example:

WHERE: ((4 * 20 + 1)^3 + (4 * 20 + 3)^3) = 1103228

THEN: ((4 * 20 + 1)^3 + (4 * 20 + 3)^3) /((4 * 31 + 1) + (4 * 09 + 3) = 6727 Odd Integer

If we now take our argument to include the altered properties for the original Base Integers, as long as they also add up to the same Summands then we will be able to duplicate the same results. Such as, (4a + 1) + (4b + 3) = (4 * (a + b) + 4), where the + 4 may be converted to (2 + 2), thus our property mutation within the Fermat Categories. For example:

IF: ((4 * 20 + 1)^3 + (4 * 20 + 3)^3) /((4 * 31 + 1) + (4 * 09 + 3)) = 6727 Odd Integer

THEN: ((4 * 20 + 2)^3 + (4 * 20 + 2)^3) /((4 * 31 + 1) + (4 * 09 + 3)) = 6724 Even Integer

With this type of weapon in our mathematical arsenal, we may now address the Goldbach Conjecture with renewed vigor.

I appreciate all your attention in this matter,

Brian S. McMillan

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P.S. Also, when 4n + 2 numbers are multiplied by any Odd Integer at all, they remain a 4n + 2 number along with its non root-able property. Thank you.

P.P.S. Furthermore, the sum of any two terms when raised to Any power greater than 2, will never yield a Prime Number, when the exponents of the two terms both share at least (one) Odd Prime as a common factor. It matters not that the exponents of the two terms ARE or ARE NOT equal. The only exception to the above is the trivial result where both terms equal 1, then obviously the result will always equal 2. Thank you.

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http://www.pumpraser.com/physics.htm