Number Theory is Impossible Without These 7 Things
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- Опубликовано: 29 сен 2024
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MATH CHALLENGE: Try to prove Fermat's Last Theorem for the case n=3, namely: there exist no positive integers a, b, c such that a^3+b^3=c^3.
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🌟 Number Theory Origins: Exploring the Fascinating World of Numbers!
📊🎓 Join us as we delve into the intriguing history and key concepts that shaped this foundational field of mathematics. Here’s a sneak peek into what you’ll discover:
1. The Beginnings with Pythagoras 📐
2. Euclid’s Algorithm 🧮
3. Modular Arithmetic 🔢
4. Fermat’s Last Theorem 💡
5. Algebraic Number Theory 🔍
6. Analytic Number Theory 📈
7. Geometric Number Theory 📏
Don’t forget to like, comment, and subscribe for more fascinating insights into the world of mathematics! 🚀✨
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1. The Beginnings with Pythagoras 📐
2. Euclid’s Algorithm 🧮
3. Modular Arithmetic 🔢
4. Fermat’s Last Theorem 💡
5. Algebraic Number Theory 🔍
6. Analytic Number Theory 📈
7. Geometric Number Theory 📏
I thought there was a simple modular arithmetic reason that x^3 + y^3 = z^3 had no solutions in the natural numbers. Unfortunately I couldn't find a contradiction this way.
The best I found for restrictions is mod 9, because it turns out that cubes mod 9 could only have remainder 0, 1, or 8. This means one of x,y,z must be divisible by 3. Without loss of generality, we can consider two cases:
1) y=0 (mod 3), x = z != 0 (mod 3)
2) z=0 (mod 3), x = - y != 0 (mod 3)
Note: y^3 = z^3 - x^3 = (z-x)(z^2 + zx + x^2)
I couldn't figure out how to go any further with this line of attack.
I’m glad to see you took on the challenge! you’re on the right track.
I’m working on a video right now where I’ll go through all the proof.
The way I found is to write the following c^3=a^3+b^3=(a+b)*(a^2-ab+b^2) , and from that conclude that (a+b) and (a^2-ab+b^2) must be perfect cubes, once we assume a,b,c in c^3=a^3+b^3 to be the “minimal solution” (thus gcd(a,b)=1). From this you can build a system of equations that will take you to a contradiction for any solution you look for (in my case I got to the contradiction that a and b have a common divisor).
Let me know if that makes sense 😎
Just subscribed. Please keep making these videos. You all deserve much more views. This content is gold
Thanks Omar! We will keep making them 😎🤙🏻 let us know what type of content you are interested in
Thank you very much sir and mam.
Love from India.
Your videos very helpful for me. I learn more things through your videos
@@OpPhilo03 that’s awesome! Thanks for the nice words. Love from Italy 😎
Andrew Wiles' persistence is unbelievable, I hope I could tackle one of the major unsolved mathematical problem.
Yeah, you are right. Which major unsolved problems would you like to tackle or that would interest you most?
@@dibeos Maybe the Birch and Swinerton-Dyer conjecture.
@@jbangz2023 ok, I honestly have no idea what it is, but I will search it right now. If you want you can explain a little bit to me right here… thanks for the suggestion anyway 😉👌🏻
That’s interesting, they don’t teach us that one algorithm to find the largest common denominator in Italy. They each us to factor the numbers and to pick all the common factors with the smallest exponent.
But I don’t think that this is taught this way only in Italy, in other parts of the world as well. The method we showed in the video though is the original one
I tried to solve the problem using the property a^3 + b^3 = (a+b)(a^2 - ab + b^2)
c^3 = (a+b)(a^2 - ab + b^2)
c = [(a+b)(a^2 - ab + b^2)]^(1/3)
c = [(a+b)((a+b)^2 -3ab)]^(1/3)
c = [(a+b)^3]^(1/3) + [(a+b)*(-3ab)]^(1/3)
c = a+ b + [-3ab(a+b)]^(1/3)
c = a + b - [3ab(a+b)]^(1/3)
3ab(a+b) has to be equal to 27k^3, with k being an integer, to c be an integer.
ab(a+b)=9k^3
I couldn't progress from there :/
Anyways, greetings from Brazil!
Number theory is so cool. I see the language of God. The programming of our Universe. The wiring under the board. Higher dimensional truth.
@@AllegraPersephone it is interesting huh?!? Number theory is, arguably, the foundation of all sciences. Would you agree with that?
@@dibeos Intuitively I feel that while math is essential for pretty much all advanced modern sciences, I would say the foundations were in philosophy. But then it is semantics about the meaning of the words. I think of early humans, maybe stone age people, or even an earlier or other hominid species, and how the heavens were their teacher. Two perfect circles in the sky. Day and night. The polar star. The wandering stars. The seasons. The measure of time. That is the true foundation of our knowledge.
@@dibeos You know in ancient times the letters of a language were also numbers, so a calculation of 2 × 3 = 6 would be B × C = F. There were two forms of this that I know of for Hebrew and Greek, and a further 2 methods where for example in Hebrew the first would be 1 to 22, and another would go from 1 to 10, and then after 10 it would go up in 10s to 100, and then in 100s to 400. The words for these as I understand them today are Gematria (Hebrew) and Isopsephy (Greek).
@@dibeos Now the Bible is full of fascinating things from number theory and math in general. Consider two foundational verses, in Genesis 1:1 and John 1:1, both start with "In the beginning". If you do a particular calculation, the exact same calculation on both of these verses, you get Pi from Genesis 1:1 and 'e' from John 1:1, both to around 99.999% accuracy. Of course I don't need to explain the significance or meaning of these numbers to you, as I usually do with others. The actual calculation is the product of each letter in the verse divided by the product of each word. What do you make of that?
@@AllegraPersephone it’s interesting! But I really need time to process all of it haha
Vi agora que você é doutor pela USP, IME ou ICMC?
Oi Vinicius! Tudo bem se respondo em inglês? (Só porque o canal é em inglês…)
I did a bachelors in physics at the University of São Paulo, Brazil, masters in mathematics at the University of Kiev, Ukraine, and (did not complete) my PhD in mathematics at the University of Udine, Italy.
@@dibeos oh, I read it wrong, but still amazing. I'm majoring in mathematics at USP, in ICMC. Great to see a fellow!
@@viniciuscilla oh yeah!!! Let me know how I can help you (maybe making videos on things that would help you somehow)
@@dibeos Considering your background, why did you change from physics to math? Personally I was really in doubt about which area to follow when I was to decide, specially since I came from 2 years in electrical engineering. I ended up chosing math, but I think many people that love STEM in general have troubles to decide.
@@viniciuscilla You know, the “unfortunate” thing is that there are too many interesting things to study in both math and physics, so it is impossible to really learn deeply any subject without sacrificing the other. I chose math because after having a strong background in physics I noticed that if I knew pure and applied math to a high enough level I would be able to learn pure math and theoretical physics. I’m very pleased with my decision, honestly. I think it’s easier to move from physics to math than the other way around. So I think you made the right decision as well haha
Subscribed
@@amiladissanayake341 Thanks Amila!!! Let us know what topics you enjoy so that we can post about them here in the channel 😎
I appreciate your videos, thank you! Would you ever be open to covering semi-rings (tropical or otherwise), or Category Theory?
Hey Alex, thanks for the encouragement! I’m definitely open to publishing videos on any subjects related to math & physics, so feel free to comment what content you’d like to see. I personally do not have much knowledge on semi-rings, but I do know some Category Theory and I think that there are many interesting insights from the field. I guarantee you we’ll make a video on it soon. Let me study a little about semi-rings and add it to our list of ideas 😎
@@dibeos Cathegory theory would be great 🥰
@@adamsmainchannel3789 thank you!! I’m convinced hahaha 😎 then, after you watch it, let us know if the video was deep enough, because it’s really hard to talk about everything having just one week to prepare it
Please do a deep dive on Category Theory @@dibeos
@@Mowrioh we will do it soon!! 😎
7 things THALA for a reason 💛💛💛💛💛💛💛
This is just an intuition. For a^2 + b^2 = c^2, which can be rewritten as (a)^2 + (b)^2 = (c)^2, there is a specific relationship for c - b (for integers) that make the equation is true. (The generalization of Pythagorean triples).
So, for a^4 + b^4 = c^4, rewritten as (a^2)^2 + (b^2)^2 = (c^2)^2, only when c=1 and b = 0 will the relationship for "c" - "b" [in this case: c^2 - b^2] be true.
Analogously, a^3 + b^3 = c^3 can be rewritten as (a^3/2)^2 + (b^3/2)^2 = (c^3/2)^2, but except for c = 1 and b=0, no integer solutions will conform to c^(3/2) - b^(3/2).
Solutions where a b or c are 0 are not allowed.
Thank you for the insightful comment 😎
Your intuition about the relationships between a^2 + b^2 = c^2 and their generalizations to higher powers is very interesting.
For a^4 + b^4 = c^4, rewriting it as (a^2)^2 + (b^2)^2 = (c^2)^2 does indeed show the structure of these equations, though as you said, this only holds trivially (c=1 and b=0).
Anyway, I think it is a very good strategy to tackle this problem by analyzing particular cases first, as you are doing.
@@dibeos Except, must something to the 4th power conform to the "shape" of the Pythagorean triples? That's the hole, I suppose. :)
Another "Stoater" as we say in Scotland.
@jonathanlister5644 I had to look it up, but if I understood correctly -- thank you 😎
Thank you for the video! Although I have worked with Modulars before in mathematics we only went over basics, I appreciate now the geometric aspects of it with regards to the visual idea of remainders. Great video Sofia and Luca!
Thanks again Riley!! It’s a pity that this video could not be “deeper”, but we are working on a video about a very cool subsection of number theory. It will be a little more technical, but I hope that it’s gonna be easy to understand for everyone anyway 😎
The square root of phi squared plus one squared equals phi squared. Every integer version of this is similar to the Fibonacci sequence. ❤
Sorry, I did not understand, but I’m genuinely curious. Can you explain?
@@dibeos howdy! So 1/1, 2/1, 3/2, 5/3 and so on is approaching phi. 3, 4, 5 and other integer right angle triangles are approaching 1, square root of phi and phi! So beautiful 🤩
@@buckleysangel7019 yeah, it is beautiful! I’m planning to make a video where I show a bunch of cool sequences that converge to the phi, e, pi, etc… do you think it would be an interesting video?! 🤔
@@dibeos I do! Here’s a fun thing I noticed. If you take six 1s and then continue a la Fibonacci, but by adding the six 1s you get six. Then add 6 to the previous 5 ones you be 11. Do digital root for 11, you get two. Combine all the ones. You get 162. A very close approximation for phi. Then do a Archimedean style hexagon. Use 1.62 for each side. 6x1.62 is 9.72. In digital roots 9 is also 0. This is a very close approximation of e! Now take the hexagon to a dodecagon. Each of the 12 sides is phi squared do to the right angle triangles. Phi squared times 12 is 31.416…. Which is a very close approximation of pi! Seems weird by moving decimals, yet this is a way of figuring out some cool approximations to some important numbers without a calculator. Thank you for the videos!
@@buckleysangel7019 wow interesting… yeah there are definitely many unexpected connections between series and “important” numbers in mathematics (phi, pi, e,…). You inspired my hahah I’ll definitely make a video about it
As mathematician in Number Theory I love this video🥰
Also I happy for mentioning Cryptography, bc I cryptographer also😍
That’s awesome! You gave me the idea hahah now I want to make another one on Number Theory, but deeper (more technical)
@@dibeos oh, it would be nice
Bc in this video you took all the history from ancient Greeks to 18 century
And maybe, I will mention that if Euclid not prove that quantity of prime numbers is infinitely many, maybe we would be not hunting for more and more ambitiously big prime numbers
And I will be write in next comment some idea for the next deep video
@@SobTim-eu3xu do it please. I think prime numbers will always be interesting because they look so “innocent”, but actually there is a world of their own to explore
@@dibeosyes, you right, if video would be full of prime numbers, it will be called cryptography is impossible without this 7 things, and this is not whole math(but algorithms can be looked by they formulas)
But prime numbers, as you say is really a big world
My idea is(not all can be in video)
Prime numbers
Riemann hypothesis
Euler phi function(Fermar little theorem, Euler theorem) solving modular equations using phi function
Kronecker/Jacobi/Legendre symbol
I think that's it
Do you have an email to send some ideas?
@@SobTim-eu3xu that’s awesome!!! I like this list so far. And yes, I do have an email: dibeos.contact@gmail.com
Bernhard Riemann was born in 1826, long after Leonhard Euler's death death in 1783, yet you credit both Euler (9:53) and Riemann (10:45) with the origin of the infinite product over primes formulation of the zeta function.
Please clarify this.
Hi Keith,
Thank you for your comment and for pointing that out. You're absolutely right-Euler and Riemann were indeed from different periods. Euler introduced the concept of the zeta function and its infinite product representation over primes. However, Riemann later expanded on Euler's work, particularly with his famous hypothesis related to the zeta function. My intention was to credit both for their significant contributions to the development of the zeta function and its properties. I will ensure to clarify this better in future content.
Thanks for watching! 😎
So you're French.
I was born in Brazil.
1. Numbers
2. ?
The 7 things we selected are:
1. Pythagorean theorem
2. Euclid’s algorithm
3. Modular arithmetic
4. Fermat’s Last Theorem
5. Algebraic Number Theory
6. Analytic Number Theory
7. Geometric Number Theory
• Pythagorean
• Remainders
• Primes
What else ?
The 7 things we selected are:
1. Pythagorean theorem
2. Euclid’s algorithm
3. Modular arithmetic
4. Fermat’s Last Theorem
5. Algebraic Number Theory
6. Analytic Number Theory
7. Geometric Number Theory
Thank you Luca and Sofia! You are great!!
:39 Pythagoras had absolutely NOTHING to do with the discovery of the Diagonal Rule, which was the name given to it by the Northern Africans (not just Egypt) and the Mesopotamians over a thousand years before Pythagoras was even born. Also let's not forget the knowledge and use of this theorem by both the Chinese and Indians well before the birth of Pythagoras.
@@williejohnson5172 interesting 🤔 , did you read that in a specific book? I’d love to learn more about it
@@dibeos This is common knowledge in most classes on the history of mathematics. A great resource and video is Norman Wilberger's You Tube series on Rational Trigonometry. He gives several lectures on the mathematics of the Mesopotamians, and the Diagonal rule and the ukulu of the Mesopotamians and the seked of the Egyptians (Kemetians).
@@williejohnson5172 that’s awesome 😎
Great video, could we have one talking about thermodynamics or complex number
Thanks for the comment, Toby&endy!! 😎 I will add both of them to the list. Actually, about complex numbers I was already preparing one. Do you have any specific requirements inside of thermodynamics and complex numbers?
Love thr vid! Very interesting topic and I learned quite alot! You deserve more subs❤
Thanks! We appreciate it, and we’ll keep on publishing. Let us know what you are interested in and what you liked specifically about the video (this way we can double on it) 😎🤙🏻
@dibeos i just liked how you two kept speaking with each other and explaining to us :) It felt like a natural conversation that I am part of. Most of my questions were asked, and you answered them all :)
@@lunarthicclipse8219 cool 😎
ok so that's equivalent to asking if there's any point of rational coordinates along the elliptic curve X^3 + Y^3 = 1 …
I need to take a class abt that lol, I remember there are things to be said, but not much more…
An elliptic curve can be expressed as (X/A)^2+(Y/B)^2=1 , where A and B can be any value. This way you can (for example) graph the ellipse and ask about the points that satisfy this equation, for different parameters (or degrees of freedom) A and B
@@dibeos I mean, I wasn't talking about the equation of an ellipse, I was talking about the elliptic curve X^3 + Y^3 = 1, if you don't see what I'm talking about it might be interesting to look it up, elliptic curves are kinda trendy nowadays… and proving Ferma's last theorem for n = 3 is asking whether there are any rational-coordinates points on the elliptic curve X^3 + Y^3 = 1.
@@m9l0m6nmelkior7 aaah ok, I see what you mean know! Yeah, probably we will make a video about it too.
@@dibeos That would be great !!
I think that curve expands further than will fit into your margins
In mathematics, why can a division with and a vector, and a division with zero not be defined?
Thank you for your question, Kisho!
In mathematics, division involving vectors is not defined in the same way as division with scalar quantities. Vectors are multidimensional quantities and division in the traditional sense does not apply to them. Instead, operations like dot product, cross product, and scalar multiplication are used to manipulate vectors.
As for division by zero, it's undefined because it leads to contradictions and mathematical inconsistencies (and discontinuities). If we could divide by zero, we would end up with results that don't make sense within our number system. For example, if a/0 = b, then a = b * 0. Since any number multiplied by zero is zero, there would be no unique solution for b, leading to indeterminate forms and inconsistencies.
I hope this answers your question! If I did not understand your questions correctly, let me know and I can answer again 😎🤙🏻
Division to be useful has to have certain properties. Division is basically just inverted multiplication, so if you wanna study division with vectors you're gonna need to study algebras over a field. It turns out that, even if you want multiplication to only be associative (not commutative, not even division) then you only have a very limited number of algebras that actually work. It also turns out that, for multiplication of real vectors to have an actual inverse, you need something called a multi vector, and then you get into geometric algebra
@@enpeacemusic192 super cool! Do you have any book or article to recommend, so that I can study this specific theme?
@@dibeos this video is really good:
ruclips.net/video/60z_hpEAtD8/видео.htmlsi=SZN9NE0sehfjy6Tt
I think most if not all stuff on that channel is about geometric algebra
@@enpeacemusic192 thanks!! 😎
..a most usefull video !!!
Thanks! Let us know what you liked exactly and what kind of content you’d like to see more often in the channel 😎
@@dibeos ...of we're possible ,abstracto álgebra would be marvelous,..your tech , your edition is really super !!
lest gooo another video
💪🏻😎
❤❤❤
Let us know what you liked and what kind of content you’d like to see in the channel, please! 😎
@@dibeos u r doing great keep going
this man moves his muscles so annoyingly
Are you talking about Wiles?
yeah (i guess), he moves like an elder man but he's young. he should work out more. i'd love that more.
Bro, that's pure hate
@@LuckyCrab_ it is
Personally, I think anyone who uses the phrase, number theory, is simply illiterate.
There are four categories of Grammar: Common Grammar, Arithmetic, Algebra and Geometry. which are differentiated in accordance with one of the four methods of binary recursion. Every grammar is simply a method of effecting binary recursion. Fact, no theory at all.
May as well say that reality is a theory.
I hate illiterate parrots.
Yeah, I understand your point. And if you use the literal definition of “theory” then the fields you mentioned are not theories, but facts as you said. However, once we understand that “Number Theory” is just a name we picked (as a convention) to describe these fields (even if this nomenclature is not the best choice in the world), then I have no problem using it. Examples of bad nomenclatures in the English language (but that work very well for their purpose: fast communication): sunset and sunrise (technically the sun is not rising or setting, but everybody knows what it means, and no one is called “illiterate” for using these words…). Same thing with Number theory. It’s a bad choice, since these are facts not theories, but calling people illiterate just because of a conventional unfortunate nomenclature seems a little extreme to me…
@@dibeos Here is another point, operations on information is common to every member of our grammar matrix, Common Grammar, Arithmetic, Algebra and Geometry.
And, as the computer demonstrates today, as Plato said, every grammar is effected by binary recursion, and as Plato, noted, Geometry can be used to produce visual paradigms for any logical process, yet we have so called mathematicians who clearly are illiterate, calling their gibberish mathematics, when MATH, are the operations perform on information, and are not subject to be claimed to have anything to do with higher, lower, or between the bread math.
So, yes, geometry can do all the so called arithmetic, algebra and common grammar computations, using none of their symbol sets.
So, instead of mincing words and making illiterate statements, why not teach mathematicians how to be literate.
Minkowski is pronounced the European way , which using English phonetics is Minkovskee, not Minkauski. The correct sound for 3.14159.. is the European , Greek , in English phonetics Pee or Pea, not the (idiotic = wrong idea ) Pai. Adolf Hitler is not Aidolf or Eidolf The A sounds as the a in March, Mars. Equally mispronounced in English is Germain, adulterated to Germaine as in Maine. The way English pronounces many European names makes a European faint or weep.
Instead of wasting 13 minutes of your life just click on the table that reads "more."
I’m sorry you didn’t like the video. We tried to make the description as loyal as possible to the content of the video. But we will keep it shorter for the next videos 😉🤙🏻 thanks for the tip!
@@dibeos The length of the video isn't my true concern, it's that people do not find genuine answers on the internet, what they do find are superficial Google searches echoed again and again and again. All the same, thanks for taking the time and energy to respond.
@@dibeos By the by, social engineering stickers are in place to do away with abstract thought. Slapping our masters digital stickers into a conversation serves no purpose whatsoever. Think for yourself and set an example next time.
@@ObsidianMonarch Did you just hate on smilies? 😂 I'm a mathematician and I LOVE smilies. Cheers
Eughhhhhh
uh-RITH-muh-tick is how you pronounce the noun "arithmetic." Its adjective is "arithmetical" and is pronounced air-rith-MET-tick-cull.
Awesome, I’m happy you like my beautiful accent 😎🤙🏻😂