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Fermat's Last Theorem plays a very interesting role in pedagogy. Today, every student can
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make the discovery by looking at a
school globe or add a wooden cube!
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Are there short and simple ways to prove Last theorem?
Do need to spend a hundred or more pages
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and help of artificial intelligence (AI) to find a prove? Beyond any doubt neural networks are
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a very useful tools for broadcasting knowledge among billions of people.
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Thus Bob and Alice will help us!
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the Fermat's Last Theorem plays a very interesting role in pedagogy. Nowadays every student can make a
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discovery by looking at a school globe or wooden Cube for child. In this video Let's consider the
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following questions: what is topology and set
theory? Why a ball and a cube are homeomorphic?
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and what homeomorphism is? We'll discuss the
irreducible conflict between form and content
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which is embedded in this theorem. Then we will
make generalizations and conclusions about the
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fundamental properties of our Universe. Begin
from the beginning. Fermat's Last Theorem was
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formulated in 1637 and it states that this
equation for the whole numbers a^n (^ - degree)
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+ b^n in degree n = c^n has no
solution for n greater than 2, except zero values
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This theorem was formulated by Pierre de Fermat in the margins of the book Arithmetic by Diaphanus
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of Alexandria in the 3rd Century A.D. Thanks to
Diophantus we treat algebraic equations as it is
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convenient and familiar to us. French mathematician
Pierre de Fermat wrote in the margin of the book
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"I have discovered a truly marvellous proof of this, which this margin is too narrow to contain".
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In the 1990s Princeton mathematician Andrew
Wiles shocked the world with a breakthrough
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standing on the shores of Number Theory. He
began dreaming of a bridge to Harmonic analysis
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they also showed that Fry's elliptic curve
can't exist which means that a solution to
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Fermast's equation can't exist either. Subsequently descendants denigrated the French mathematician
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and believed that he made a lightweight judgment simply put bragging a nd lying.
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I'm Andrew Granville I work in analytic number Theory when I was young
I worked on formats Last Theorem before it was
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proved today I work in ideas of L function you
cannot prove them because you cannot go back to
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anything more primitive than those propositions
themselves they are not in need of justification
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this is particularly famous example by Peter
schultzer he had a very very difficult proof
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but he wasn't 100 sure of he would try and say
well this is true and you should know this lead
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and me would say I don't know this at all
you're gonna have to explain this better
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the basis for such an assertion was a hundred
plus page proof by Andrew Wiles prepared by him
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in 1994. the question arise of their short and
simple ways to prove the Fermat's Last Theorem
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to understand this is it necessary to waste
100 or more pages and help of artificial
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intelligence we refer to 500 page work of Shinichi Mochizuki Japanese mathematician working in Number
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Theory and arithmetic geometry from Kyoto
University dealing with the "ABC hypothesis"
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in number Theory. But there is another approach
Minhyong Kim, a mathematician at the University
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of Oxford said "it should be possible to use
ideas from physicists to solve problems in
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Number Theory but we haven't thought carefully
enough about how to set up such a framework"
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and one more his quotation: "We're at a point
where our understanding of physics is mature
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enough and there are enough number theorists
interested in it to make a push"
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To the posed question about short and simple ways to prove the Fermat's Last Theorem the answer is "Yes!"
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And this method is associated with a mental
experiment. Consider a construction of 3
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concentric nested n-cubes or balls with centers
at the origin with edges or radii just equal to
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natural numbers a, b, c. Why these whole numbers which are mentioned in the Last Theorem will
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necessarily be different? Suppose the opposite. If the first two terms and are equal to each other
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then in this case it is easy to see that the root
of 2 must be represented as a fraction say p and q
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Let's reduce p and q by greatest common divisor
and they become mutually prime numbers but then
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we go to this equation and make sure that we
have an even number on the right and left sides
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For instance if degree n equal to 2 the right
part is divisible by 4 because here p^2
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is the result at least twice prime number two so
divide by four the right part of equation and make
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sure that the left part must also be divisible by
2 and again by 2. Thus p and q become even and
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we have a ride at a contradiction. For the case
of degrees higher than 2 the proof is quite
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similar so it is no less general to put that
whole numbers a < b which in turn is
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less than c. Three years ago in 2020
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I formulated a proof of Fermat's Last Theorem which is that if you take the 3 n-cubes here symbolically depicted on the plane a-Small cube you layer
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around it a certain number of unit cubes, create
the b- Middle cube layer another certain number of
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layers create c-Large cube where volume of a small cube is equals to volume of difference of c-Large
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and b-Middle cubes. It is easy to make sure that in the project of "A house for a capricious rock star"
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the condition of equivalence of the volumes of
the Studio and the Winter Garden excludes central
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symmetry and vice versa and you get a construction that doesn't really exist in nature for the case
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n > 2. It's amazing! You will never
be able to put each unit cube 1^n from a-Small
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n-cube in correspondence with another unit cube of this subset of layers between the Middle and the
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Large one in such a way that you do not destroy
the symmetry of the construction and do not allow
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voids. Generally speaking these subsets are not
equivalent to each other this is the main idea of
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the proof. This is due to the fact that this figure
has the property of central symmetry and does not
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contain inhomogeneities as a result each layer
in this figure is not comparable to another layer.
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Thus each layer in this figure is not reducible
to another layer here additivity conditions and
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Axiom of Measure do not work. In other words it
is impossible to speak about addition of volumes.
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Why did Andrew Wiles need over a hundred more pages to prove it, if half of page is enough or
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six faces of wooden cube for child? Rospatent
put me "guilty of plagiarism" with Andrew Wiles
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proof for which he received an able award in 2016.
Rospatent refused the State registration of a
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patent 2021501435 for an industrial design under the pretext of "violating public morality". Rospatents claim
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about "plagiarism" of the proof does not stand up to any criticism because the proof you see is
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original new and it really breaks the stereotypes
formed in science about the absence of a short
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proof of Fermat's Last Theorem. Let us imagine
that we have concentric balls nested within
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each other ball with the natural radii a, b,
c that we are looking for we will distinguish
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between a ball and a sphere that encompasses the ball the sphere is the so-called "layer" for the
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ball so the above notations are adopted here here is the ball and here is the sphere a layer it has
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Dimension one unit less n-1. This is clear from physics and math courses remember the formulas for the
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length of a circle 2πR and the area of a circle πR^2 the area of a sphere and the volume of a ball. It is
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interesting to imagine that in the one-dimensional world we would have such an open ball in the form
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of a segment excluding its endpoints or Zero
dimensional spheres located at a distance R from
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the origin of coordinate. Now for the case of the 2-dimensional plane the open ball becomes the familiar circle.
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Excluding the 1-dimensional sphere already
considered finally for the three-dimensional
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case we have the usual ball resembling a soccer
ball it is closed by a two-dimensional sphere we
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again exclude an arbitrary meridian from the
sphere as a result we got acquainted with the
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sequence of geometrical elements on the sphere
of dimension from 1 up to n minus one these are
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the so-called "hypermeridians" which will be
useful for us in the future let's continue
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the experiment with the cascade of spheres this is the encompassing meridian let's try to change the
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step for example what happened and this is the
code that allows us to create this design it's
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based on an understanding of set theory now let's
look at the basics of set theory and make a small
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generalization if a triple of natural numbers a b
c exists C existence quanta for a brief notation
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then in this case we can map every point of space
from this interval between the Middle Ball and the
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Big Ball into a Small Ball and since this whole
construction is symmetric this means that every
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sphere that surrounds these spheres can be mapped
into many other spheres. The Fermat's Last Theorem
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states that this is impossible, why? - Because that's the physics of our Universe! It manifests itself
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in the topological properties of figures let's
talk about continuous functions and relations
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that work in topology. Topology is the science
of geometric objects that modify these objects
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in the most arbitrary way but nothing breaks
and all transformations are reversible here you
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see the Russian figure skater Kamilla Valeeva she is plastic, she performs a beautiful smooth dance.
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Let's imagine a certain function F mapping the set X to the set Y similarly you can see how Kamilla