Fermat's Last Theorem: Exploring x^n + y^n = z^n and Beyond

  • Thread starter Russell E. Rierson
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In summary, the conversation starts with the equation x^n + y^n = z^n, which is a general identity for primes and numbers. It then introduces the concepts of multiplication, division, addition, and subtraction, and reminds us to follow the order of operations. The conversation then moves on to discuss a metric space and its distance function, and how it is related to real numbers and a manifold M. The topic of homeomorphism is brought up, but there are some missing variables and definitions that make it difficult to fully understand. The conversation then shifts to a discussion of velocity and the Heisenberg uncertainty principle, and ends with a mention of the law of excluded middle and the equation for spacetime given by John Nash. The speaker
  • #36
Hawking writes:

Information about the quantum states in a region of spacetime may be somehow coded on the boundary of the region, which has two dimensions less. This is like the way that a hologram carries a three dimensional image on a two dimensional surface.

So information is encoded on the 2-dimensional boundary of spacetime.


http://www.encyclopedia4u.com/t/topological-space.html [Broken]



Any metric space turns into a topological space if define the open sets to be generated by the set of all open balls. This includes useful infinite-dimensional spaces like Banach spaces and Hilbert spaces studied in functional analysis.

http://www.encyclopedia4u.com/m/metric-space.html [Broken]

Metric space

A metric space is a space where a distance between points is defined. It is a topological space.

Formally, a metric space is a set of points M with an associated distance function (also called a metric) d : M × M -> R (where R is the set of real numbers). For all x, y, z in M, this function is required to satisfy the following conditions:


d(x, y) ≥ 0
d(x, x) = 0 (reflexivity)
if d(x, y) = 0 then x = y (identity of indiscernibles)
d(x, y) = d(y, x) (symmetry)
d(x, z) ≤ d(x, y) + d(y, z) (triangle inequality).

These axioms express intuitive notions about the concept of "distance": distances between different spots are positive and the distance between x and y is the same as the distance between y and x. The triangle inequality means that if you go from x to z directly, that is no longer than going first from x to y, and then from y to z. In Euclidean geometry, this is easy to see. Metric spaces allow this concept to be extended to a more abstract setting.
A metric space in which every Cauchy sequence has a limit is said to be complete.

Metric spaces are paracompact Hausdorff spaces and hence normal (indeed they are perfectly normal). An important consequence is that every metric space admits partitions of unity and that every continuous real-valued function defined on a closed subset of a metric space can be extended to a continuous map on the whole space (Tietze extension theorem). It is also true that every real-valued Lipschitz-continuous map defined on a subset of a metric space can be extended to a Lipschitz-continuous map on the whole space.

Two metric spaces (M1, d1) and (M2, d2) are called isometrically isomorphic iff there exists a bijective function f : M1 → M2 with the property d2(f(x), f(y)) = d1(x, y) for all x, y in M1. In this case, the two spaces are essentially identical. An isometry is a function f with the stated property, which is then necessarily injective but may fail to be surjective.

Every metric space is isometrically isomorphic to a closed subset of some normed vector space. Every complete metric space is isometrically isomorphic to a closed subset of some Banach space.


Distance between points and sets
If (M,d) is a metric space, S is a subset of M and x is a point of M, we define the distance from x to S as

d(x,S) = inf {d(x,s) : s ∈ S}
Then d(x, S) = 0 if and only if x belongs to the closure of S. Furthermore, we have the following generalization of the triangle inequality:
d(x,S) ≤ d(x,y) + d(y,S)
which in particular shows that the map x |-> d(x,S) is continuous.

 
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  • #37
so is it that the known universe is four dimensional with a 2 dimensional boundary? why are there 11 dimensions?
 
  • #38
Originally posted by phoenixthoth
so is it that the known universe is four dimensional with a 2 dimensional boundary? why are there 11 dimensions?

[zz)] [zz)] [zz)]

http://superstringtheory.com/basics/basic6a.html


T duality
The duality symmetry that obscures our ability to distinguish between large and small distance scales is called T-duality, and comes about from the compactification of extra space dimensions in a ten dimensional superstring theory. Let's take the X9 direction in flat ten-dimensional spacetime, and compactify it into a circle of radius R, so that



A particle traveling around this circle will have its momentum quantized in integer multiples of 1/R, and a particle in the nth quantized momentum state will contribute to the total mass squared of the particle as



A string can travel around the circle, too, and the contribution to the string mass squared is the same as above.
But a closed string can also wrap around the circle, something a particle cannot do. The number of times the string winds around the circle is called the winding number, denoted as w below, and w is also quantized in integer units. Tension is energy per unit length, and the the wrapped string has energy from being stretched around the circular dimension. The winding contribution Ew to the string energy is equal to the string tension Tstring times the total length of the wrapped string, which is the circumference of the circle multiplied by the number of times w that the string is wrapped around the circle.



where



tells us the length scale Ls of string theory.
The total mass squared for each mode of the closed string is



The integers N and Ñ are the number of oscillation modes excited on a closed string in the right-moving and left-moving directions around the string.
The above formula is invariant under the exchange



In other words, we can exchange compactification radius R with radius a'/R if we exchange the winding modes with the quantized momentum modes.
This mode exchange is the basis of the duality known as T-duality. Notice that if the compactification radius R is much smaller than the string scale Ls, then the compactification radius after the winding and momentum modes are exchanged is much larger than the string scale Ls. So T-duality obscures the difference between compactified dimensions that are much bigger than the string scale, and those that are much smaller than the string scale.
T-duality relates type IIA superstring theory to type IIB superstring theory, and it relates heterotic SO(32) superstring theory to heterotic E8XE8 superstring theory. Notice that a duality relationship between IIA and IIB theory is very unexpected, because type IIA theory has massless fermions of both chiralities, making it a non-chiral theory, whereas type IIB theory is a chiral theory and has massless fermions with only a single chirality.
T-duality is something unique to string physics. It's something point particles cannot do, because they don't have winding modes. If string theory is a correct theory of Nature, then this implies that on some deep level, the separation between large vs. small distance scales in physics is not a fixed separation but a fluid one, dependent upon the type of probe we use to measure distance, and how we count the states of the probe.
This sounds like it goes against all traditional physics, but this is indeed a reasonable outcome for a quantum theory of gravity, because gravity comes from the metric tensor field that tells us the distances between events in spacetime.




 
  • #39
Take two opposing mirrors as an overly simplistic example.

The number of self contained images increase, while the size of the images decrease.

[<-[->[<-->]<-]->]


A self including set.
 
  • #40
Originally posted by Russell E. Rierson
Take two opposing mirrors as an overly simplistic example.

The number of self contained images increase, while the size of the images decrease.

[<-[->[<-->]<-]->]


A self including set.


You are correct in this:Information about the quantum states in a region of spacetime may be somehow coded on the boundary of the region, which has two dimensions less. This is like the way that a hologram carries a three dimensional image on a two dimensional surface.


The thing is this:Take two opposing mirrors as an overly simplistic example.

The number of self contained images increase, while the size of the images decrease.

[<-[->[<-->]<-]->]


A self including set.

Has allready been contemplated here:http://groups.msn.com/Youcanseehomefromhere [Broken]

And the Universe has 2-Dimensional locations where there is an intersection of dimensional projection.

We are used to our conceptions that our 3-D world is continuous and surrounds the micro-quantum fields, the question is which surrounds what and where. The space between Galaxies is made from 2-Dimensional fields, which surround a 3-D space containing our Galaxy, the action of Dimensionality constricts our Galaxy, the Galaxy 3-D is Embedded in a 2-D lattice, quite a Quantum Leap in perceptional thinking is needed to see this (dont you think!).

The Flat Geometric space between Galaxies is not the same generic make up as the Field-Space that is in and around Matter(Atoms).

When we look inwards from our 3-D space down to Matter, 3-Dimensions allways Surround the (QUANTUM-2-D and inner Vaccum/fields) matter we are observing. When we turn our observation around and look outwards into the Universe at Large, we are SURROUNDED by a 2-Dimensional Field(E-M-Vaccum), quite the reverse of our pre-conceptual Spacetime idealistic Galactic Notions.
 
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  • #41
Take two opposing mirrors as an overly simplistic example.

The number of self contained images increase, while the size of the images decrease.

[<-[->[<-->]<-]->]


A self including set, all relations are "intrinsic". The "outside" = first iteration = alpha = 0. The final "last" iteration = omega = 0.

alpha = omega



http://superstringtheory.com/basics/basic6a.html



T duality

The duality symmetry that obscures our ability to distinguish between large and small distance scales is called T-duality,




If space is *quantized* yet also continuous, then it too, has the property called "wave-particle" duality. If space consists of indivisible units, then a measurement of space means that Fermat's last theorem holds, for it.

According to the Pythagorean theorem:

x^2 + y^2 = z^2

All possible integer solutions are then rerpresented as:

[a^2 - b^2]^2 + [2ab]^2 = [a^2 + b^2]^2

a^4 -2(ab)^2 + b^4 + 4(ab)^2 =

a^4 + 2(ab)^2 + b^4 = [a^2 + b^2]^2




all odd numbers can be represented as:

[a^2 - b^2] or Z^p - Y^p

if Y is an "even" natural n and Z is odd, same for a and b .

Fermat's last theorem, for integers a,b,Z,Y,p:

[a^2 - b^2]^p + Y^p = Z^p

[a^2 - b^2]^p = Z^p - Y^p

a^2 - b^2 = [Z^p - Y^p]^[1/p]

When Z^p - Y^p is a prime number, it cannot have an integer root.

a^2 - b^2 is not an integer, for [Z^p - Y^p]^[1/p] , for a,b,Z,Y,p, unless p = 2.


To every set A and every condition S(x) there corresponds a set B whose elements are exactly those elements x of A for which S(x) holds. This is the axiom that leads to Russell's paradox. For if we let the condition S(x) be: not(x element of x), then B = {x in A such that x is not in x}. Is B a member of B? If it is, then it isn't; and if it isn't, then it is. Therefore B cannot be in A, meaning that nothing contains everything.

This means that relativity holds in the "topological" sense and T-duality is correct.



Vector fields can be defined on manifolds in a concise "algebraic" way. Given an arbitrary vector field on R^n, the directional derivative of function f may be formed in the direction of vector v, denoted by vf.

x^1, ... , x^n can represent coordinates on R^ n[though eventually coordinate independence is preferred].

The "Einstein summation convention"

vf = v^1 @f/@x^1 + ... +v^n @f/@x^n

where @ denotes the partial derivative symbol.

Infinitely many differentiable functions may then be defined on manifold M


f+g = g+f

f+ [g+h] = [f+g]+h

f[gh] = [fg]h

f[g+h] = fg+fh

[f+g]h = fh+gh

1f = f

Vector Fields:

v[f+g] = v[f] + v[g]

v[af] = av[f]

v[fg] = v[f]g + fv[g]

The tangent vector space is infinitesimally distrubuted over M.


If memory serves, the distance formula for two dimensions:

ds^2 = g_11 dx^2 + 2g_12 dydx + g_22 dy^2
 
  • #42
A + B = C

A^2 + B^2 = C + A^2 - A + B^2 - B

A^2 + B^2 = [A+B] + A[A-1] + B[B-1]

A^p + B^p = [A+B] + A*[A^(p-1)-1] + B*[B^(p-1)-1]



3^2 + 4^2 = 5^2

3^3 + 4^3 = 5^3 - 34

34 is a Fibonacci number.

I wonder if the Fibonacci series can lead to a "proof" of Fermat's Last Theorem?

1,2,3,5,8,13,21,34,55,89,144,233,377,610,...


5^2 - 3^2 = 2^4

[2^2 - 1^2]^2 + [2*2*1]^2 = [2^2 + 1^2]^2

5^2 + [2*2*3]^2 = 13^2



Squares of fibonacci numbers give fibonacci numbers or multiples of them:


1^2 + 2^2 = 5

2^2 + 3^2 = 13

3^2 + 5^2 = 34

5^2 + 8^2 = 89

8^2 + 13^2 = 233

But Fibonacci cubes aren't so predictable?

1^3 + 2^3 = 9 = 8 + 1

2^3 + 3^3 = 35 = 34 + 1

3^3 + 5^3 = 152 = 144 + 8

5^3 + 8^3 = 637 = 1 + 5 + 21 + 610


These Fermat characters are HILARIOUS!


http://www.fermatproof.com/

Is the "proof" valid?

He appears to be using the rule of Pythagorean triples, trying to extend it out to all values of n > 2. Then using reducto ad absurdum to establish the absurdity of n > 2



Interesting diagrams with the right triangles

...2
...22
...222
..1111

= 4^2

2+2+2+2+2+2+1+1+1+1 = 4^2

6+4 = 10

1+2+3+4

10 + 5 = 15

10*2 + 5*1 = 5^2

15 + 6 = 21

15*2 + 6*1 = 6^2


N(N+1)/2 = 1+2+3+...+ N

Interesting...


Z = [X^p + Y^p]/[Z^(p-1)]

For p > 2, Z cannot be an integer...



That is the big question, why do two cubes not result in another cube?

3^2 + 4^2 = 6[4]+1

5^2 + 12^2 = 14[12]+1

7^2 + 24^2 = 26[24]+1

and

3^3 + 4^3 + 5^3 = 43[5]+1 = 6^3

3^3 + 4^3 = [45/2]*[4] +1






http://www.umcs.maine.edu/~chaitin/unknowable/ch1.html [Broken]

At that same time that Turing shows that any formalism for reasoning is incomplete, he exhibits a universal formalism for computing: the machine language of Turing machines. At the same time that he gives us a better proof of Gödel's incompleteness theorem, he gives us a way out. Hilbert's mistake was to advocate artificial languages for carrying out proofs. This doesn't work because of incompleteness, because of the fact that every formal axiomatic system is limited in power. But that's not the case with artificial languages for expressing algorithms. Because computational universality, the fact that almost any computer programming language can express all possible algorithms, is actually a very important form of completeness! It's the theoretical basis for the entire computer industry!


[...]


And I begin to get the idea that maybe I can borrow a mysterious idea from physics and use it in metamathematics, the idea of randomness! I begin to suspect that perhaps sometimes the reason that mathematicians can't figure out what's going on is because nothing is going on, because there is no structure, there is no mathematical pattern to be discovered. Randomness is where reason stops, it's a statement that things are accidental, meaningless, unpredictable, & happen for no reason.


Interesting...


The "ABC conjecture" :

http://www.maa.org/mathland/mathtrek_12_8.html [Broken]
 
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  • #43
whenever people raise questions you can't answer or don't understand you post 'bored' smilies. Perhaps you are a genius. Perhaps you're an idiotic crank. I know where my money is
 
  • #44
Originally posted by matt grime
whenever people raise questions you can't answer or don't understand you post 'bored' smilies. Perhaps you are a genius. Perhaps you're an idiotic crank. I know where my money is

That appears to be a "bi-valent" logic statement.

What if I am not a genius and I am not a crank?




I could say that you have no sense of humor or that you are

But I would probably be wrong[I hope] [?]
 
  • #45
Originally posted by Russell E. Rierson
In simplest terms, a hypersurface is an [ n-1] dimensional space. For example, 3 dimensional space can be generalized to a 2-dimensional surface.

What the h**l is the relevance of R^17 ? 17 legs to the right triangle? All that is required is R^3 + 1 Then the [n-1] hypersurface is 2D.

x^n + y^n = z^n

You were attempting to define a well-defined algebraic operation on an arbitrary manifold, ie turning it into field or division algebra, at least that was one interpretation and you didn't present any explanation of what you wre doing. No such structure exists fo R^{17}, as most mathematicians know (division algebra structures exist for 2,4,8,16 degree extensions of R and no others), so where does your Fermat Last Theorem argument get you here? Your last 'sentence' is numerically challenged.

If you are neither an idiot nor a crank I'd lose that money. But then I just dislike people posting huge quoted bits of text for neither rhyme nor reason and not justifying any of their bizarre assertions, so I might be letting it affect my judgement.
 
  • #46

as most mathematicians know (division algebra structures exist for 2,4,8,16 degree extensions of R and no others)


n-1 = 2


Information may be stored on the two dimensional boundary of space, much like a holographic image.

Space is a perception of separation between points, perception being cognitive processing of the "mind".

Now we come to the "Universal Set", which is analogous to a Universal Algorithm.

Algorithms contain information.

The abstract[algorithm] contains the concrete[units-bits of information]

Quite simple really, the "Platonic form" is an abstract description of a concrete object or space[topology].

The Platonic form is isomorphic to the concrete instantiation.

P = Platonic form

C = Concrete object or space

P<---->C

P[C] = C[P]
 
  • #47
while it may be correct, can you give like some information about the isomorphism to further prove that the two realms are isomorphic? like in what context are they isomorphic (as sets, as groups, as vector spaces, etc.)?
 
  • #48
Originally posted by phoenixthoth
while it may be correct, can you give like some information about the isomorphism to further prove that the two realms are isomorphic? like in what context are they isomorphic (as sets, as groups, as vector spaces, etc.)?

Well phoenix, I have been trying to explain, much to the chagrin
of

An algorithm contains the information with regards to the necessary and sufficient topologies-geometries, RELATIONS, and other "bijections" that are required. Algorithms correspond to configurations.

I recall you posted the equation f[x] = x

Say F = U , where F is a particular calculation that feeds back into itself.

F = U

G[F] = U

U[...H[G[F]]] = U

A Platonic form is an abstract "thought" form, or configuration, where if it has the necessary internally self consistent "logics" to sustain its own existence, then it actually "exists". The Platonic thought form is also known as the Schrodinger wave function. A damped oscillation.

As I explained a couple of posts back, while being rudely interrupted and insulted, the isomorphism is analogous to two opposing mirrors, where the images are intersecting.

[<-[->[<-[-><-]->]<-]->]

The real physics of the above diagram corresponds as a one to one and onto correspondence, which creates a condition of standing-wave resonance. The optimal configuration is in accordance with the "action principle". The axiom of choice is governed by an operator that forces it to "choose" the most optimal configuration.

So I know that you are strictly a mathematician, but my advice is to grapple with the physics also, if you are going to "prove" the Absolute Infinity...

The information density of the universal system must be increasing. The increase of information density is analogous to a pressure gradient.

[density 1]--->[density 2]--->[density 3]---> ... --->[density n]

The universal laws of nature are explained in terms of symmetry identities. The completed infinities, mathematician Georg Cantor's infinite sets, could be explained as cardinal identities, akin to "qualia" [Universally distributed attributes] distributing over their universes in accordance with a guiding principle.

Completed infinities, called "alephs" are distributive in nature, similar to the way that a set of "red" objects has the distributive property of redness. Properties, or "attributes" like red are numbers in the sense that they interact algebraically according to algebraic laws. Take one object away from the set of red objects and the distributive identity "red" still describes the set. The distributive identity[attribute] "natural number" or "real number" describes an entire collection of individual objects.

The alephs can be set into a one to one correspondence with a proper subset of of themselves.

On the other hand, a finite set of objects becomes less than its previous self, if an element is removed from it.


[abstract representation]-->[semantic mapping]-->[represented system]


An abstract representation is exactly that, "abstract". It is not a space, or time, but is instead a product of consciousness, or a mental construct.

On one level of stratification, two photons are separate. On another level, of stratification, the photons have zero separation.

Instantaneous communication between two objects, separated by a distance interval, is equivalent to zero separation[zero boundary] between the two objects.

According to the book "Gravitation", chapter 15, geometry of spacetime gives instructions to matter telling matter to follow the straightest path, which is a geodesic. Matter in turn, tells spacetime geometry how to curve in such a way, as to guarantee the conservation of momentum and energy. The Einstein tensor[geometric feature-description] is also conserved in this relationship between matter and the spacetime geometry. Eli Cartan's "boundary of a boundary equals zero."

The Topological spaces are defined as being diffeomorphism invariant. Intersecting cotangent bundles[manifolds] are the set of all possible configurations of a system, i.e. they describe the phase space of the system. They have "zero" boundary...

Waves are then abstract distributions and particles are convergent "concrete" localizations.


Here is what Max Tegmark says:

http://www.sciam.com/article.cfm?articleID=000F1EDD-B48A-1E90-8EA5809EC5880000&pageNumber=7&catID=2





As a way out of this conundrum, I have suggested that complete mathematical symmetry holds: that all mathematical structures exist physically as well. Every mathematical structure corresponds to a parallel universe. The elements of this multiverse do not reside in the same space but exist outside of space and time. Most of them are probably devoid of observers. This hypothesis can be viewed as a form of radical Platonism, asserting that the mathematical structures in Plato's realm of ideas or the "mindscape" of mathematician Rudy Rucker of San Jose State University exist in a physical sense. It is akin to what cosmologist John D. Barrow of the University of Cambridge refers to as "ð in the sky," what the late Harvard University philosopher Robert Nozick called the principle of fecundity and what the late Princeton philosopher David K. Lewis called modal realism. Level IV brings closure to the hierarchy of multiverses, because any self-consistent fundamental physical theory can be phrased as some kind of mathematical structure.


I certainly hope that you understand the "relevance" of the above quote...







Scientific objectivity for "Plato":

http://www.mmsysgrp.com/stefanik.htm

Quote:

Thesis: Scientific objectivity is best characterized by the concept of invariance as explicated in category theory than the concept of truth as explicated in mathematical logic.
 
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  • #49
You know I was reading through this thread, tepidly amused while I happened to read my coffee cup and Russell's theory began make more and more sense.

What did the coffee cup say?

If you can't dazzle them with brilliance, bury them in bull****.
 
  • #50
Fermat's Last Theorem

That solution to Fermat's Last Theorem is my father's work.

My father (photo included) was an electrical engineer who worked at Vought on rocket missile systems. He was really smart and I have total faith in his theorem www.fermatproof.com . Please forward this link to interested parties.

Thanks!
 
  • #51
Why not put the Fermat equation in terms of one variable, where the polynomial must have positive integer roots, for degree n = prime ?

f[x] + g[x] = h[x]

f[x] = x^p

g[x] = [x+y]^p

h[x] = [x+z]^p

where y and z are arbitrary integer constants > 0.


f[x]+g[x] = h[x]


1 + g[x]/f[x] = h[x]/f[x]


{f[x]g'[x] - g[x]f '[x] }/{f[x]}^2 =

{f[x]h'[x] - h[x]f '[x] }/{f[x]}^2


divide out the {f [x]}^2

rearrange terms

f[x]{g'[x] - h'[x]} = f '[x]{g[x] - h[x]}

f[x]/f '[x] = {g[x] - h[x]}/{g'[x] - h'[x]}

f[x]/f '[x] also equals:

{h[x] - g[x]}/{h'[x] - g'[x]}


[1.] x^p + y^p = z^p

differentiate [1.]

p*x^[p-1] * x' + p*y^[p-1] * y' = p*z^[p-1] * z'

divide both sides by p

[2.] x^[p-1] * x' + y^[p-1] * y' = z^[p-1] * z'

[1.] - [2.] & factor

x^[p-1] *[xy'-yx'] = z^[p-1] *[zy'- yz']

[xy' - yx'] = z^[p-1] *[zy'-yz']/x^[p-1]

Since x,y,z are relatively prime, z^[p-1] does not divide x^[p-1]

interesting...
 
  • #52
non-formally differentiating functions of integers is it now?
 
  • #53
Originally posted by matt grime
non-formally differentiating functions of integers is it now?

No, in the last paragraph, x,y,z are polynomial functions of x.
 
  • #54
Types are entities built up in a finite number of steps, beginning with an arbitrarily chosen object, 0, also identifiable with the number 0, in accordance to certain rules.

[1.]

0 is a type

[2.]

If n is a positive integer and t_1,...,t_n are types then the sequence [t_1,...,t_n] is a type. Types can be assigned to all individuals, sets, and relations, as needed, with regards to the universe under consideration.

[3.]

Let the set of all types be T.

If, for every type t, B_t, contains all relations of A_t, then the higher order structure, M, which is any set, B_t, that is indexed in T( or basically function defined on T ), such, that for any non-empty set of individuals A_t, where B_t is a subset of A_t, with every element of A_t appearing as a function satisfying certain necessary and sufficient conditions, then the higher order structure M, is full.

If no B_t contains any relation repeatedly, such that the value a function, f, takes on each value at most "once", then the higher order structure is normal. Loosely speaking, a structure is full and normal if B_t = A_t for all t.

At the very rock bottom of the set theory universe, is the singular entity called the "empty set" At first there is nothing at all, with possibly an undefined, unknown, realm in the reverse direction of the empty set, such that the imagination runs amok contemplating the possibilities.

The whole universe of set theory is constructed from the empty set, and since it is the raw idea of set formation, the set theory universe in ever increasing levels of complexity.

The empty set is the simplest set, which is the set that has no elements, represented as { }

The next most elementary set is the set containing the empty set, i.e. { {} }

etc...etc...etc...


A topological group G is a topological space, and also a group, such, that group multiplication ab = c is a continuous function from G x G into G and the inverse operation a^-1 = b is also a continuous function in a topological space. Yes, it is much to the bewilderment and chagrin of certain friends at physics forums. So one of the defining characteristics of a topological space, is, that if ab = c and W is an open neighborhood of c, then there exist open neighborhoods U and V of a and b respectively, such that UV is a subset of W.
 
  • #55
Pythagorean triples:

http://grail.cba.csuohio.edu/~somos/rtritab.txt

The Fibonacci numbers are also within Pythagorean triples

a^2 + b^2 = c^2 .

15^2 + 8^2 = 17^2

35^2 +12^2 = 37^2

99^2 + 20^2 = 101^2

255^2 + 32^2 = 257^2

675^2 + 52^2 = 677^2

1763^2 + 84^2 = 1765^2

4623^2 + 136^2 = 4625^2

etc.

At closer inspection:

(4^2-1)^2 + (2*4)^2 = (4^2+1)^2

(6^2-1)^2 + (2*6)^2 = (6^2+1)^2

(10^2-1)^2 + (2*10)^2 = (10^2+1)^2

(16^2-1)^2 + (2*16)^2 = (16^2+1)^2

(26^2-1)^2 + (2*26)^2 = (26^2+1)^2

(42^2-1)^2 + (2*42)^2 = (42^2+1)^2

(68^2-1)^2 + (2*68)^2 = (68^2+1)^2

etc...

Notice:

4 = 2*2

6 = 2*3

10 = 2*5

16 = 2*8

26 = 2*13

42 = 2*21

68 = 2*34

2*Fibonacci

etc...

Interesting...




While researching "squares" of numbers I find that ratios of certain types of squares always involve something like a + (2b)^(1/2).

I found many squares that have successive ratios that tend towards the number
3 + (8)^(1/2)

This is of course one solution to the polynomial x^2 - 6x + 1 = 0.

The challenge is to answer the question "why"
are these squares ratios tending towards
3 + (8)^(1/2) as the numbers are getting larger?

I will give some "lists" of these squares!

3^2 + 4^2 = 5^2

20^2 + 21^2 = 29^2

119^2 + 120^2 = 169^2

696^2 + 697^2 = 985^2

4059^2 + 4060^2 = 5741^2

23660^2 + 23661^2 = 33461^2

137903^2 + 137904^2 = 195025^2

Now the ratios

29/5 = 5.8

169/29 = 5.827586207...

985/169 = 5.828402367...

5741/985 = 5.828426396...

33461/5741 = 5.828427103...

195025/33461 = 5.828427124...

3 + (8)^(1/2) = 5.828427125...

This number also appears for the successive ratios of any of the three numbers
(a, b, c), a2/a1, b2/b1, or c2/c1 of the Pythagorean triples on the above list.

Here are some more...

3 = 2^2 - 1

17 = 4^2 + 1

99 = 10^2 - 1

577 = 24^2 + 1

3363 = 58^2 - 1

19601 = 140^2 + 1

114243 = 338^2 - 1

665857 = 816^2 + 1

The ratios

17/3 = 5.666666667...

99/17 = 5.823529412...

577/99 = 5.828282828...

3363/577 = 5.828422877...

19601/3363 = 5.828427...

114243/19601 = 5.828427121...

665857/114243 = 5.828427125...

Also approaching 3 + (8)^(1/2)




The mathematicians tell me "no-problemo" an easy problem!

This is what one mathematician explained to me...




Hi, this is easy to explain.

Notice that your equation is basically of the form

a^2 + (a+1)^2 + 1 = b^2

Never mind, for the moment, that you are considering consecutive ratios of b. First start by trying to understand which a's and b's can be solutions to this equation.

Well, this equation is the same as:

2a^2 + 2a + 1 - b^2 = 0.

Which is the same as

4a^2 + 4a + 2 - 2b^2 = 0

which is

(2a+1)^2 -2b^2 = -1

This is a special case of Pell's Equation

X^2 - 2Y^2 = -1 (i.e. where X=2a+1 and Y=b).

Now observe that (X,Y) = (1,1) is a solution.

It turns out that all the solutions
(X_k, Y_k) to this equation are given by

X_k + sqrt(2) Y_k = (1+sqrt(2))^k where k is odd.

Now your question relates to those X_k which are odd. Now compute some examples with k = 1,3,5,7,9,11,13... you will see a pattern. This explains your observations.




1+sqrt2)^2 = (3 + sqrt8)

(1+sqrt2)^3 = (7 + sqrt50)

(1+sqrt2)^4 = (17 + sqrt288)

(1+sqrt2)^5 = (41 + sqrt1682)

(1+sqrt2)^6 = (99 + sqrt9800)

(1+sqrt2)^7 = (239 + sqrt57122)

(1+sqrt2)^8 = (577 + sqrt332928)


The Fibonacci sequence:

1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144,
233...

1+0 = 1

1+1 = 2

1+2 = 3

2+3 = 5

3+5 = 8

5+8 = 13

8+13 = 21

etc...

Fibonacci squares... also generate Fibonacci numbers...

1^2 - 0^2 = 1

1^2 + 1^2 = 2

2^2 - 1^2 = 3

2^2 + 1^2 = 5

3^2 - 1^2 = 8

3^2 + 2^2 = 13

5^2 - 2^2 = 21

5^2 + 3^2 = 34

8^2 - 3^2 = 55

8^2 + 5^2 = 89

13^2 - 5^2 = 144

13^2 + 8^2 = 233

21^2 -8^2 = 377

21^2 + 13^2 = 610

etc...

The consecutive ratios of these numbers... are of course approaching (1 + sqrt(5))/2

The "golden" ratio...

Sum the digits of 2^n and get a repeating pattern:

2^1 = 2

2^2 = 4

2^3 = 8

2^4 = 16, 1+6 = 7

2^5 = 32, 3+2 = 5

2^6 = 64, 6+4 = 10, 1+0 = 1

2^7 ---> 2

2^8 ---> 4

2^9 ---> 8

2^10 ---> 7

2^11 ---> 5

2^12 ---> 1

2^13 ---> 2

2^14 ---> 4

etc...

etc...

etc...
 
<h2>What is Fermat's Last Theorem?</h2><p>Fermat's Last Theorem is a mathematical theorem proposed by French mathematician Pierre de Fermat in the 17th century. It states that no three positive integers a, b, and c can satisfy the equation an + bn = cn for any integer value of n greater than 2.</p><h2>Why is Fermat's Last Theorem important?</h2><p>Fermat's Last Theorem is important because it is one of the most famous and long-standing unsolved problems in mathematics. It has intrigued and challenged mathematicians for centuries, and its proof required the development of new mathematical concepts and techniques.</p><h2>What is the significance of x^n + y^n = z^n in Fermat's Last Theorem?</h2><p>The equation x^n + y^n = z^n is the specific form of the equation an + bn = cn that is addressed in Fermat's Last Theorem. It is also known as the "Fermat equation" and is used to represent the theorem in a more general form.</p><h2>Has Fermat's Last Theorem been proven?</h2><p>Yes, in 1994, British mathematician Andrew Wiles presented a proof for Fermat's Last Theorem after working on it for seven years. His proof was later reviewed and accepted by the mathematical community, making it one of the most celebrated achievements in mathematics.</p><h2>What are some applications of Fermat's Last Theorem?</h2><p>Fermat's Last Theorem has no direct practical applications, but its proof has led to advancements in various mathematical fields, such as algebraic number theory and elliptic curves. It has also inspired further research and has shown the power and beauty of mathematics.</p>

What is Fermat's Last Theorem?

Fermat's Last Theorem is a mathematical theorem proposed by French mathematician Pierre de Fermat in the 17th century. It states that no three positive integers a, b, and c can satisfy the equation an + bn = cn for any integer value of n greater than 2.

Why is Fermat's Last Theorem important?

Fermat's Last Theorem is important because it is one of the most famous and long-standing unsolved problems in mathematics. It has intrigued and challenged mathematicians for centuries, and its proof required the development of new mathematical concepts and techniques.

What is the significance of x^n + y^n = z^n in Fermat's Last Theorem?

The equation x^n + y^n = z^n is the specific form of the equation an + bn = cn that is addressed in Fermat's Last Theorem. It is also known as the "Fermat equation" and is used to represent the theorem in a more general form.

Has Fermat's Last Theorem been proven?

Yes, in 1994, British mathematician Andrew Wiles presented a proof for Fermat's Last Theorem after working on it for seven years. His proof was later reviewed and accepted by the mathematical community, making it one of the most celebrated achievements in mathematics.

What are some applications of Fermat's Last Theorem?

Fermat's Last Theorem has no direct practical applications, but its proof has led to advancements in various mathematical fields, such as algebraic number theory and elliptic curves. It has also inspired further research and has shown the power and beauty of mathematics.

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