# Thread: Fermat's Theorem, Relativity, Quantum Field Theory

1. ## Fermat's Theorem, Relativity, Quantum Field Theory

Proof of Fermat's Theorem

I will be tracking (most) updates to this proof here

Updates to Proof of Fermat's Last Theorem

(The is an "on the fly" update; make sure your cache is empty to get the latest version; even then, there may be some lag.)

If Fermat's expression were true (), , , and would be a Pythagorean Triple. They are not a Pythagorean Triple by the Binomial Theorem, where rem(a,b,n) > 0.

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

See post #81 for diagram.

QED

The dates in my signature reflect the most recent updates - and they are still a work in progress.

This is the link to my proof of Fermat's Theorem. Some in this thread bray "Nonsense" without context like jackasses - only Grapes seems to read documents and critique in context (occasionally), alerting me to possible algebraic errors, which are much appreciated for what they are (I'm writing a lot of tex and have no proofreader). However, as of this date (2/24/2017) he has yet to learn that triangles are expressed in two dimensions, rather than one, and therefore the significance of the dot product in Pythagorean triangles, not to mention the more general cases of circles, squares, and volumes.

Neither Neverfly nor The Naked Emperor have any idea WTF I am talking about as will be clear from their posts ....

The context of the proof is that for two generalized variables, two dimensions are required for mathematical (i.e., arithmetical) operations on real numbers as well as integers. This is because the dimensions have to be related by a common metric (i.e., must touch, cannot be parallel in two dimensions), so there is a common connection (the "zero" point, which defines the origin of each dimension as the midpoint of all possible lengths for that dimension). Thus, two dimensions must have a common origin in order to compare metrics.

This is true for real numbers, as well as integers, which is why the Binomial Theorem is applicable - Pythagorean Triplets begin with orthogonal dimensions, and the case for powers is expanded via the Binomial Expansion.

If there is no such connection, it can be removed by "imaginary" (no pun intended); i.e. complex numbers, which is an operation fundamental to the Pauli characterization of "spin" via the and matrices which treats number lines on either side of the origin as independent (1,-1), expanded by Dirac to the complete description of the circle (1,-1,1,-1).

The Relativistic Unit Circle shows that the first integer for positive definite integers is created as when an arbitrary metric is applied via the so-called "time dilation" equation.

Quantum Field Theory then applies the result to two dimensions (1,1) where for positive definite quantities equally and oppositely directed in terms of the null vector, like the analysis above, where the interaction in terms of n and h can be identified with 2ab from the Binomial Expansion for n=2.

The derivative then becomes a function of first dividing out the successive interactions given by (a+b), and then taking the limit as b ->0, so that multiplied by the null vector. (if a=0, there is no integer to begin with). When this is accomplished, the constant a can be identified with the slope of a straight line in the coordinate relation y = Ax, a = A, where the derivatives can then be identified with the Jacobian of the metric tensor in GTR.

The constant can be identified as the "mass" of a particular photon relative to the "zero point" energy at which the photon has no perceived mass (e.g., the parking lot on earth, unless you get a sunburn). The "c" is identified as a specific value (t=1) from the "measured" displacement current from Maxwell's equation, via the force constants from Ampere's and Coulomb's laws.

The process of setting h and n equal to zero in the QFT characterization removes interactions from the theory to the two dimensional characterization of Dirac for two photons (photon -equivalent particles by deBroglie), and finally two a single photon by setting so the energy of the single photon is given by where c has a different value for each photon depending on its relation to the zero point energy.

Probabilistically, that energy is usually given by a Gaussian spread around some reference black-body temperature in the parking lot. The context can be expanded to the solar system, the galaxy, and beyond if one uses one's fertile imagination. May the Force be with you.. (it ain't with me, as far as I know...

2. ## Re: Fermat's Theorem, Relativity, Quantum Field Theory

are you going completely nuts and just throwing everything, including the kitchen sink, into the mix?

3. ## Re: Fermat's Theorem, Relativity, Quantum Field Theory

I have revised the OP to reflect a corrected error, and its relevance to STR, GTR, QFT, and Foundations of Mathematics.

4. ## Re: Fermat's Theorem, Relativity, Quantum Field Theory

Still same old errors of trying to use binomial.

5. ## Re: Fermat's Theorem, Relativity, Quantum Field Theory

Originally Posted by BuleriaChk
( Proof of Fermat's Theorem PDF revision - updated)

I wrote yesterday that I had discovered a problem in the most recent version of my signature document. It has now been fixed, and clarifies the fundamental issue- the distinction between covariance and contra-variance. It had no effect on the proof of Fermat's Theorem, but does clarify the distinction between GTR and QFT, and results in a proof of Quantum Triviality re. the Higgs Boson... (if there is a god, he doesn't speak to us...

For a linear system there is no distinction, and the ratio of is preserved in the relativistic unit circle, since , so the energy is always . For QFT this means setting invariant, and letting vary. so that represents a change in density/unit length and represents a change in length per unit density for the selected (invariant) values of and .

.
To preserve
in a coordinate system, one sets either
OR , but not both in the deconstruction of ; the first (contra-variant) alternative is normally used in QFT ( in the unit circle); I had mistakenly used the second (covariant) alternative . In SRT, for , so the distinction is moot for an unperturbed system.

(For Fermat's proof think , , and as invariants; which immediately translates to the Binomial Theorem for the case and then to Fermat's theorem for with a and b in separate dimensions (number lines).

For on the same number line (one dimension) the integers refer to the same unique number, so the only possible solutions are or .

For the contravariant case in the RUC, , , ; I had inadvertently used , and so was inconsistent.

However the sections where are correct, and form the foundation of QFT, where the specification of spin ( ) in terms of or depending on the context, but spin is also directly related to of the Binomial Theorem for , which is shown in the Dirac formulation for a circle representing two “bare” particles and their interaction where , in two dimensions.

This is a shorthand description of my mistake and is correction in context; there is much more to say, but most of it is already in documents I have already written in the course of this journey, which began when I decided to try to learn STR, GTR, and QFT on retirement. he document is online again; I have eliminated redundancy, and am continuing to proof it as I go along. And yes, it does apply to Goedel's proof, Foundations of Mathematics, etc. etc. as well)

------------------

Proof of Fermat's Theorem

For Fermat's Theorem it is always possible to arrange the three integers in the equation where and (corresponding to the Axiom of Choice).

Then dividing each side by results in .

If then so that .

If then so that .

If neither nor , then AND , so that neither can be an integer ( cannot be a lowest common denominator; i.e. unless the integers form a Pythagorean Triple). Therefore which is the case for the Binomial Expansion for , where , .

Note that if is not an integer, then cannot be an integer so cannot be an integer; neither can kc for any integer k, so cannot be an integer.

The case for can easily be extended for , since , where , and only disappears if or , as in the one-dimensional case.

Then QED (i.e., Fermat's Theorem is valid).

This result can be extended to GTR where the non-linear terms in rem(a,b,n) can be shown to correspond to the Christoffel curvature symbols, which relates STR, GTR, The Binomial Theorem and Fermat's Theorem to show that the classical Lagrangian for a two-particle sysem cannot conserve energy for n>2 (and not even for n=2 if spin is included).

This can be extended through the Euler Conjecture for many-body particles (integers), since if Fermat's Theorem is true for two particles, multiple particles can be arranged by the Axiom of Choice in a similar fashion (computer simulation notwithstanding...)

???

Your "proof" of Fermat's Last Theorem is a mess.

Originally Posted by BuleriaChk
------------------

Proof of Fermat's Theorem

For Fermat's Theorem it is always possible to arrange the three integers in the equation where and (corresponding to the Axiom of Choice).
Has nothing to do with the Axiom of Choice.
Then dividing each side by results in .

If then so that .

If then so that .

If neither nor , then AND , so that neither can be an integer ( cannot be a lowest common denominator; i.e. unless the integers form a Pythagorean Triple).
You did start off by assuming , so they are a Pythagorean Triple.

But, of course a/c and b/c are not integers, because they're less than 1
Therefore
You assumed that . So they are a Pythagorean Triple, and the equality holds. Your "proof" seems to say that Pythagorean Triples are not Pythagroean Triples. What a mess.
which is the case for the Binomial Expansion for , where , .

Note that if is not an integer,
You assumed that c was an integer. Now your "proof" says that if c is an integer, then it's not an integer. What a mess.
then cannot be an integer so cannot be an integer; neither can kc for any integer k, so cannot be an integer.

The case for can easily be extended for , since , where , and only disappears if or , as in the one-dimensional case.

Then QED (i.e., Fermat's Theorem is valid).

6. ## Re: Fermat's Theorem, Relativity, Quantum Field Theory

I have revised my pdf Fermat's Last Theorem in the OP to include the fact that the scalar coefficients of vectors in the space (a,b) commute even without the real metric comparison, which means there is the interaction product 2ab even in the case of Pythagorean triplets, which makes them completely consistent with the Binomial Expansion for the case n=2 and the results of the Dirac formulation in Quantum Field Theory.

My signature pdf has also been updated to include this analysis with reference, but I have removed most of the material relating to Fermat there (except for the link), so it is more focused on the relation of the Binomial Expansion to Quantum Field Theory and GTR...

In all cases, refer to the pdf's from the OP, since they are the latest update of my work. There is not much point in making the effort to write tex redundantly here. If you think you have a real criticism, pm me.

(I do make algebraic mistakes on occasion, since it is difficult for me to keep the overall conceptual ideas clear.)

7. ## Re: Fermat's Theorem, Relativity, Quantum Field Theory

Originally Posted by BuleriaChk
I have revised my pdf Fermat's Last Theorem in the OP to include the fact that the scalar coefficients of vectors in the space (a,b) commute even without the real metric comparison, which means there is the interaction product 2ab even in the case of Pythagorean triplets, which makes them completely consistent with the Binomial Expansion for the case n=2 and the results of the Dirac formulation in Quantum Field Theory.

My signature pdf has also been updated to include this analysis with reference, but I have removed most of the material relating to Fermat there (except for the link), so it is more focused on the relation of the Binomial Expansion to Quantum Field Theory and GTR...

In all cases, refer to the pdf's from the OP, since they are the latest update of my work. There is not much point in making the effort to write tex redundantly here. If you think you have a real criticism, pm me.
Kinda unbelievable
Originally Posted by grapes

8. ## Re: Fermat's Theorem, Relativity, Quantum Field Theory

Tried the algebra again. I think it is correct now, but I get nightmares about this stuff. I believe the analysis is correct. I'll continue to revisit it.

The analysis provides an alternative definition to the derivative (that John Gabriel was attempting) in terms of linear system theory.
(That's why Dirac and Pauli got the big bucks....

The point is that the basis of the relativistic unit circle's energy in one dimension is , which is the first integer with a single basis in one dimension (the basis interacts with itself). For two integers (bases), the energy of the circle is if there is no interaction (so just raw "counting" applies subjectively - 2ab=0). For higher dimensions (integers) the diagram is no longer a circle, but the interactions are handled in terms of quantum field theory, where the initial state is invariant for the interpretations.

However, if there is a common origin, then Pauli and Dirac introduce parity as independent variables, which give the (equivalent) resultant interaction of for a = b = 1.

The analysis therefore doesn't involve c until the very end, since the interaction is created by the (hypothetical) linearity of (a,b) if there is a common origin (the vectors are not "affine" -i.e., parallel but different in one dimension). That was the source of Einstein's Parallel transport" to try to introduce gravity (or god) as an additional force to that of electromagnetism interpreted as c (the foundation of observation).

9. ## Re: Fermat's Theorem, Relativity, Quantum Field Theory

Malaria, do you even know what the axiom of choice is?

10. ## Re: Fermat's Theorem, Relativity, Quantum Field Theory

Once more into the breach - yet another shot at the algebra in the PDF. I really think it is ok now. (the Proof is still valid)

As an exercise, set a = b = 1 in the Expansion

for a model of the basis set (1,1) in two dimensions...

Page 1 of 29 12311 ... Last

#### Posting Permissions

• You may not post new threads
• You may not post replies
• You may not post attachments
• You may not edit your posts
•