The Heart of Fermat's Last Theorem - Numberphile
video description
This is similar to the Galois Theory proof that you can't square a circle. Using construction, you can start with length one and create line segments of irrational length. For example the square root of two is pretty easy to construct. You can create an infinite number of different irrational lengths, like seven plus the square root of two divided by two. But you can't create ALL irrational lengths. Like the dark square bishop, you can get lots of places within the rules of the game, but you can't get everywhere. The proof shows that squaring the circle involves the wrong sort of irrational numbers -- the square root of pi I think, but that is just detail.
The video seems to be suggesting that similar restrictions apply to curves and modular forms. If I understand correctly, the Fermat equations look like elliptic equations, but they represent places on the playing field that the rules of modular forms don't allow.
Date: 2022-04-08
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Comments and reviews: 9
quint
i took an attempt at fermats last theorem as an engineer and i started with a-2+b-2 = c-2 and i figured out that you can create a square made out of 8 triangles thats length b and width a and at the center there was a smaller square so the entire square is 7 - 7. it turns out that the formula for c can be written out as c = sqrt( (2 - Volume of a-b) + (-a+b-)-2 ) e. g sqrt(2- 3 - 4) + 1) = 25. anyway i thought that if in 2d space u can perceive a-2+b-2 = c-2 as a square than it may follow that in 3d space u can make a cube following the same rules. so i tried making a cube using 6 sides that follow the rule and it turns out that no matter how you do it the cube always ends up with 4 a-2+b-2 = c-2 sides and 2 sides that are oneven rectangles instead of squares. and so in conclusion i think you can prove fermats last theorem because the geometry in 3d space isnt even and this is because the legs of the triangles have to be adjacent to a different length for the cube to actually be geometrical
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i took an attempt at fermats last theorem as an engineer and i started with a-2+b-2 = c-2 and i figured out that you can create a square made out of 8 triangles thats length b and width a and at the center there was a smaller square so the entire square is 7 - 7. it turns out that the formula for c can be written out as c = sqrt( (2 - Volume of a-b) + (-a+b-)-2 ) e. g sqrt(2- 3 - 4) + 1) = 25. anyway i thought that if in 2d space u can perceive a-2+b-2 = c-2 as a square than it may follow that in 3d space u can make a cube following the same rules. so i tried making a cube using 6 sides that follow the rule and it turns out that no matter how you do it the cube always ends up with 4 a-2+b-2 = c-2 sides and 2 sides that are oneven rectangles instead of squares. and so in conclusion i think you can prove fermats last theorem because the geometry in 3d space isnt even and this is because the legs of the triangles have to be adjacent to a different length for the cube to actually be geometrical
reply
Ал-тай
Fermat's theorem-
I proved on 09/14/2016 the ONLY POSSIBLE proof of the Great Fermat's Theorem (Fermata. -
I can pronounce the formula for the proof of Fermath's great theorem: -
1 - Fermath's great theorem NEVER! and nobody! NOT! HAS BEEN PROVEN! -
2 - proven! THE ONLY POSSIBLE proof of Fermat's theorem-
3 - Fermath's great theorem is proved universally-proven for all numbers-
4 - Fermath's great theorem is proven in the requirements of himself! Fermata 1637 y. -
5 - Fermath's great theorem proved in 2 pages of a notebook-
6 - Fermath's great theorem is proved in the apparatus of Diophantus arithmetic-
7 - the proof of the great Fermath theorem, as well as the formulation, is easy for a student of the 5th grade of the school to understand! -
8 - Me! opened the GREAT! A GREAT Mystery! Fermath's theorem! (not -simple- - -mechanical- proof)-
! - NO ONE! and NEVER! (except ME! . of course) and FOR NOTHING! NOT! will find a valid proof of the FGT!
reply
Fermat's theorem-
I proved on 09/14/2016 the ONLY POSSIBLE proof of the Great Fermat's Theorem (Fermata. -
I can pronounce the formula for the proof of Fermath's great theorem: -
1 - Fermath's great theorem NEVER! and nobody! NOT! HAS BEEN PROVEN! -
2 - proven! THE ONLY POSSIBLE proof of Fermat's theorem-
3 - Fermath's great theorem is proved universally-proven for all numbers-
4 - Fermath's great theorem is proven in the requirements of himself! Fermata 1637 y. -
5 - Fermath's great theorem proved in 2 pages of a notebook-
6 - Fermath's great theorem is proved in the apparatus of Diophantus arithmetic-
7 - the proof of the great Fermath theorem, as well as the formulation, is easy for a student of the 5th grade of the school to understand! -
8 - Me! opened the GREAT! A GREAT Mystery! Fermath's theorem! (not -simple- - -mechanical- proof)-
! - NO ONE! and NEVER! (except ME! . of course) and FOR NOTHING! NOT! will find a valid proof of the FGT!
reply
Ал-тай
I have proved on 09/14/2016 the ONLY POSSIBLE proof of the Great Fermat's Theorem (Fermata. -
I can pronounce the formula for the proof of Fermat's great theorem: -
1 - Fermath's great theorem NEVER! and nobody! NOT! HAS BEEN PROVEN! and NEVER! -
2 - proven! THE ONLY POSSIBLE proof of Fermat's theorem-
2A - Me opened: - EXIST THE ONLY POSSIBLE proof of Fermat's Great Theorem-
3 - Fermat's great theorem is proved universally-proven for all numbers-
4 - Fermat's great theorem is proven in the requirements of himself! Fermata 1637 y. -
5 - Fermat's great theorem proved in 2 pages of a notebook-
6 - Fermat's great theorem is proved in the apparatus of Diophantus arithmetic-
7 - the proof of the great Fermat theorem, as well as the formulation, is easy for a student of the 5th grade of the school to understand! -
8 - Me! opened the GREAT! Mystery! Fermat's theorem! (not -simple- - -mechanical- proof)-
! - NO ONE! and NEVER!
reply
I have proved on 09/14/2016 the ONLY POSSIBLE proof of the Great Fermat's Theorem (Fermata. -
I can pronounce the formula for the proof of Fermat's great theorem: -
1 - Fermath's great theorem NEVER! and nobody! NOT! HAS BEEN PROVEN! and NEVER! -
2 - proven! THE ONLY POSSIBLE proof of Fermat's theorem-
2A - Me opened: - EXIST THE ONLY POSSIBLE proof of Fermat's Great Theorem-
3 - Fermat's great theorem is proved universally-proven for all numbers-
4 - Fermat's great theorem is proven in the requirements of himself! Fermata 1637 y. -
5 - Fermat's great theorem proved in 2 pages of a notebook-
6 - Fermat's great theorem is proved in the apparatus of Diophantus arithmetic-
7 - the proof of the great Fermat theorem, as well as the formulation, is easy for a student of the 5th grade of the school to understand! -
8 - Me! opened the GREAT! Mystery! Fermat's theorem! (not -simple- - -mechanical- proof)-
! - NO ONE! and NEVER!
reply
darkmath100
Fermat's proof was slightly bigger than the margin of a book he was reading while he came up with it. Let's say it's 2 pages long. Andrew Wiles proof for the same problem is HUNDREDS of pages long. It's not the same proof. In fact Andrew Wile unwittingly has proven that modern mathematics is hopelessly stuck on itself to the point of being pointless.
As far as I'm concerned Fermat's Theorem has yet to be solved. Andrew Wile should be ashamed of himself for putting his ego in front of mathematical simplicity. If Einstein said mathematics is the art of thinking simply then what Andrew Wile came up with isn't mathematics.
; -)
reply
Fermat's proof was slightly bigger than the margin of a book he was reading while he came up with it. Let's say it's 2 pages long. Andrew Wiles proof for the same problem is HUNDREDS of pages long. It's not the same proof. In fact Andrew Wile unwittingly has proven that modern mathematics is hopelessly stuck on itself to the point of being pointless.
As far as I'm concerned Fermat's Theorem has yet to be solved. Andrew Wile should be ashamed of himself for putting his ego in front of mathematical simplicity. If Einstein said mathematics is the art of thinking simply then what Andrew Wile came up with isn't mathematics.
; -)
reply
enlong
simple explanation of Taniyama-Shimura-Weil conjecture by 3-n/2-2 - 2 = m-2 turn into 2-3 - (3-t)-2 = (2-m)-2 (-m have symmetry with m at x-axis) have elliptic form x-3 - ax = y-2 for x = 2, a = 3-t, n=2-t, n is positive nature number, have integer solution for m at 1, 4, 7, 10. and 2, 5, 8, 11. all of multiple of 3 to infinity for modular form, 3 and 2 could be any prime number, if it have solution for m at start(m=1 or 2 for 3(=1+2, m=2 or 5 for 7(=2+5, etc)must be modular form, 2-d for d>2 is not elliptic curve any more, 2 could be any prime numbers.
reply
simple explanation of Taniyama-Shimura-Weil conjecture by 3-n/2-2 - 2 = m-2 turn into 2-3 - (3-t)-2 = (2-m)-2 (-m have symmetry with m at x-axis) have elliptic form x-3 - ax = y-2 for x = 2, a = 3-t, n=2-t, n is positive nature number, have integer solution for m at 1, 4, 7, 10. and 2, 5, 8, 11. all of multiple of 3 to infinity for modular form, 3 and 2 could be any prime number, if it have solution for m at start(m=1 or 2 for 3(=1+2, m=2 or 5 for 7(=2+5, etc)must be modular form, 2-d for d>2 is not elliptic curve any more, 2 could be any prime numbers.
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Phyein
This is exactly the kind of math vid I look for - I'm a long way from being able to understand advanced math but I love attempts to paint an intuitive analog (difficult as it may be on it's own as well) to extremely technical stuff that seems so alien. Nobody can just look at a wikipedia summary of modular forms and elliptic curves and even begin to get an idea like one that is conveyed here for example. This is the kind of thing I want to get out of learning more and more advanced math.
reply
This is exactly the kind of math vid I look for - I'm a long way from being able to understand advanced math but I love attempts to paint an intuitive analog (difficult as it may be on it's own as well) to extremely technical stuff that seems so alien. Nobody can just look at a wikipedia summary of modular forms and elliptic curves and even begin to get an idea like one that is conveyed here for example. This is the kind of thing I want to get out of learning more and more advanced math.
reply
-----
Fermat's last theorem is a fact.
(no solutions) to the equation in whole numbers for all n big from 2
3-10 +3-9
3-12. 0023143684277=
(no solutions to the equation) in whole numbers for all n big from 2
3-6. 471273626960361 +3-9
3-10=
The full proof can be found in my English books in amazon
yehuda bitton's equations 1
Yehuda Bitton's Equations 2
x=9 y=10
z=12. 0023143684277 n=3
and
x=9 y=6. 471273626960361
z=10 n=3
reply
Fermat's last theorem is a fact.
(no solutions) to the equation in whole numbers for all n big from 2
3-10 +3-9
3-12. 0023143684277=
(no solutions to the equation) in whole numbers for all n big from 2
3-6. 471273626960361 +3-9
3-10=
The full proof can be found in my English books in amazon
yehuda bitton's equations 1
Yehuda Bitton's Equations 2
x=9 y=10
z=12. 0023143684277 n=3
and
x=9 y=6. 471273626960361
z=10 n=3
reply
Denis
I got that the proof entailed merging, or at least connecting, two different branches of mathematics and how one branch can soft of be represented in another. Also the proof hinged on a contradiction. Finally, the proof is actually very complicated and probably is best done by following the argument as trying to visualise what is going on is too mindbending. Is that the feeling I was supposed to get?
I liked this video.
reply
I got that the proof entailed merging, or at least connecting, two different branches of mathematics and how one branch can soft of be represented in another. Also the proof hinged on a contradiction. Finally, the proof is actually very complicated and probably is best done by following the argument as trying to visualise what is going on is too mindbending. Is that the feeling I was supposed to get?
I liked this video.
reply
Russell
So is the slinky a representation of the modularity theorem?
And who decides when a proof is a proof and what sort of exclusions, substitutions, omissions, and limitations are permitted in developing one?
Is there any connection between patterns observed in the natural world (in flora, fauna, patterns on the surface of water, etc) and the modularity theorem?
reply
So is the slinky a representation of the modularity theorem?
And who decides when a proof is a proof and what sort of exclusions, substitutions, omissions, and limitations are permitted in developing one?
Is there any connection between patterns observed in the natural world (in flora, fauna, patterns on the surface of water, etc) and the modularity theorem?
reply
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