Awesome!! Actually I was thinking to publish of proof of the irrationality of sqrt(2) in a few weeks. Let me know if it is correct after I post it 😎

Yes!! The fact that they are coprime implies that each must be perfect cubes. Good job!

University of São Paulo (Brazil), University of Kyiv (Ukraine) and University of Udine (Italy). Sofia studied at Northumbria University (UK)

That's not the only mistake, but clearly it's the biggest one, also in the challenge he left, if you take b = q = 3, p can be anything such that q^3 != 0 modulo p^3, so the statement is perhaps incomplete

Idk what makes me think that he left it as a challenge because he couldnt prove it or he might have proven it but it contradicted his proof but he wont simply show it

@@vectorial6467 Well, after some playing around, I found out that if you take b = 3t^3 and q = 3t^2, where t is a random integer, 3b^2 = q^3 mod any p^3, so the statement is flat out wrong Maybe if you also impose that (b, q) = 1 as they must be in the problem, that would resolve the issue Edit: it doesn't solve the problem, just choose b = 3t^3, q = 3t^2 + p^3, for any integer t and all the conditions are satisfied, but q^3 is not 0 mod p^3, let's take for example t = 1 and p = 2, this means b = 3 and q = 11, in this case (p, q) = (p, b) = (q, b) = 1 as in the problem, and 3b^2 = 27 which mod 8 is 3, also q^3 = 1331 which mod 8 is also 3 so 3b^2 = q^3 mod p^3, but q^3 is not 0 mod p^3. The statement is plain wrong any meaningful condition we try to add

You are right. What I actually meant is that using the minimality argument, a and b are chosen to be coprime, so the conclusion is the same and the rest of the proof follows from it. Thanks for pointing it out.

Without even appealing to the minimal condition, note that in the specific scenario where we have a³+b³=c³ and we assume gcd(a,b,c)=1, then indeed if gcd(a,b)>1 we have a contradiction. Because if gcd(a,b)=d>1 and p|d a prime, then we have p|a and p|b. Hence p|a³+b³=c³. Since p is prime, ww must have p|c. But this implies p|gcd(a,b,c), which we assumed was 1

@@fotisp6293 oh yeah, that’s a good point!! 😎 thanks, it makes more sense

In the proof, if (e_1, e_3, …) are not multiples of 3, then their presence in the factorization cannot account for making a perfect cube. This choice simplifies the argument by ensuring that any exponent not divisible by 3 immediately leads to a contradiction.

@@dibeos yeah but why did you assume them to be exactly 3, why not 3k1, 3k3, ...?

@@dibeos I'm ignoring this part, though the proof has flaws, the fact that if gcd (a, b) = 1, and ab = x³ => a = m³, b= n³ is true. The part that's most problematic is that mod 3 part. I can't find a way to fix that

In his prove he have showed that a and -b is multiples of three, that means 3,-3 is congruence mod 3 and all 3k and -3m, where k, m is positive integer If he show that a, b is not in equal class of congruence, for example 3 and -5 3 mod 3=0 -5mod3=(1-6) mod3=1 Also I can get -5mod by this: -5+3=-2 -2+3=1 The I can conclude that 3 and -5 is not in equal class of congruence, and that means that they is not congruence mod 3, that's the point

@@SobTim-eu3xu he showed that a == -b (mod 3), from here one can write: a = 3m + r -b = 3n + r But he wrote: a = 3m + r b = 3n + r How did he conclude that? That was absolutely illogical.

If you’re using == to mean congruence a==-b mod 3 means a mod 3 has a remainder of -b which means that plus 3 should bring it up above 0 less than 3 or 0 itself. Using a negative remainder is essentially useless because if you add the modulus it should still have a valid remainder less than the modulus, almost by definition I’m afraid.

Wait you are right suppose that a=1 and b=2 , a==1(mod3) and -b=-2==1mod(3) this is true but a==1mod(3) and b==2mod(3) different remainders , so if a and -b has the same remainder when devided by 3 this doesnt mean that a and b also has the same remainder

@@vectorial6467 In your example -2==1(mod3), where 1 is the remainder and is the same remainder. b + (1)(mod3) = a

Thanks for commenting. Yes, d should be prime. You are right.

I was born in Brazil.

Yeah, this was just the first one, we want to make another video going deeper on the subject (especially showing all its applications in physics and engineering)

Thanks Agustín! Let us know what topics you’d like us to post videos about 😎

Yeah, it would be really cool because a sonic boom has a very sharp and sudden change in pressure, so it kind of looks like a step function. When applying the Fourier transform to such a signal, I imagine we get many different frequencies, which in practice would form a continuous spectrum

@@dibeos yeah, and Sonic boom not just reassemble sawtooth it has even more complex small sawtooth add in it depending on geometry. I actually wonder if there is any sonic boom that we can feel but not heard due to ultra and infra sonic sound waves.

@@user-tr4oz9cj6p I’m pretty sure that we can feel but not hear some really intense sonic booms, it's very plausible. Infrasonic and ultrasonic waves can have physical effects that we can sense even if they fall outside our hearing range. The vibrations might be felt in the form of pressure changes or vibrations

@@user-tr4oz9cj6p I’m pretty sure that we can feel but not hear some really intense sonic booms, it's very plausible. Infrasonic and ultrasonic waves can have physical effects that we can sense even if they fall outside our hearing range. The vibrations might be felt in the form of pressure changes or vibrations

Thank you!! Missed your encouraging comments haha we are happy to improve more and more each day. We are focusing on build the channel based on the intersection between what we join and what viewers are interested in watching!! Hopefully we will keep on going for many years to come 😎

Nice! We’re glad you liked it! 😎🤙🏻

yeah the subject becomes way intresting when you know its history

@@hardwork3199 i think the history is important because it gives you perspective of where the subject came from, and consequently you understand better how we got to where we are today and (most importantly) where we are heading to

Hi NaNaNa… No, we are not. I was born in Brazil but have Italian origins in my family. Sofia was born in Ukraine and spent most of her life in California and Moscow. We now live in Udine, Italy 😎

@@dibeos one of my friend looks like you and he's Italian. That's why i asked this question.

@@NaN_000 yeah, your friend must be really good looking 😎🤣

yeah he looks very italian

@@dibeos yeah lol

Yeah, I don’t know… but I think you should consider some quantum effects in your calculations

​@@dibeos Hmm is it: 0 + 1 = 1 or is it 1 + 0 = 1 or is it both simultaneously? They both have the same result, but they're not the same expression... Is this a result of Super Position, or Quantum Entanglement. They both hold to the Additive Identity and the Commutative Identity... Except one is non generative and the other is generative. Hmm... interesting. Then again, 0 and 1 are Orthogonal, Perpendicular, Normal to each other just as the Sine and Cosine functions are. Sine starts at (0,0) and Cosine starts at (0,1) and the Sine is an ODD function, and the Cosine is an Even function, but other than that, they are basically the same waveform function. They have the same waveform (shape), they have the same range and domain, they have the same period. Their only differences are their initial positions, and it is their initial positions that give them their ODD and EVEN properties. Also, they are literally horizontal translations of each other by you guessed it, 90 degrees or PI/2 radians. This is one of the reasons why I like FFTs. Linear Algebra and Vector Calculus. It's all built off of the properties of 0, 1, and the operators (+) and (=). The ability to count, add, enumerate, to apply a transformation, a translation.

Thanks Elizabeth! Nice comments like that encourage us to keep going 💪🏻 let us know what kind of math or physics videos you’d like to watch on the channel 😎

Nice, would you like us to make a video about it? What EXACTLY about it would you enjoy? 😎

@@dibeos It doesn't matter, if it involves FFTs I'm game!

Not yet… what do you think? Would it be a good idea?

🔥😎🤙🏻 C’mon Jon

Thanks for the nice comment (again)!!! It really encourages us!! I think you’ll like the video we will publish on Wednesday (I guess, from your comments)

@@dibeos OK, I will wait)

Hi Julian, thanks for your comment! We’re glad you're finding it helpful. You understood the point correctly. In the context of Fourier series, the idea isn't limited to handling just finite discontinuities. What really matters is the measure of the set of discontinuities. When we say a set has "measure zero," we mean that the total "length" of these discontinuities is negligible in comparison to the entire interval we're examining. Even if there are infinitely many discontinuities, as long as their combined measure is zero, the Fourier series can still effectively represent the function. Of course, there is a mathematical proof for this claim. This concept comes from set theory and helps in managing the problematic jumps (discontinuities) in functions like the square wave. Essentially, because these jumps take up no significant "space," the Fourier series can approximate the function almost everywhere else perfectly. Let us know if you have any more questions or need further explanation, feel free to ask 😎

Thanks! We are doing great. Hopefully people will like this video. Fourier series are veeeery interesting because it connects many areas of math and physics

Yessir! I just added it to the list of next videos. I need to be honest, I don’t know it, but I’ll take the next weeks to research and learn lots of it 😎🤙🏻

I (Luca) am a mathematician, and Sofia is a historian. Let’s say that we combine our backgrounds in order to transform technical (mathematical and physical) concepts into a well structured, and easy to follow, presentation. The power of storytelling really facilitates the learning process, as you can see in our videos 😎

You always have a positive comment to encourage us! Thanks a lot, we appreciate it!!! ❤️😎

haha i love your guys' videos and you guys are very wholesome 😊

Do you like topology? We were thinking about making another video on a specific subject inside topology. Let us know if you’d prefer that or another area of Mathematics (or Physics) 😎🤙🏻

But some errors in this video Group chat "crytography" And I will be named this video Not cryptography number theory, l like just say Cryptography I hope you can understand me, no offense

This video is great DH(Diffie-Helman) RSA ECC(Elliptic Curve Cryptography) The bestest choices And I get it that you not show calculations for ECC, but you can use the fact that you choose point P on the curve F And for each multiple of P you demonstrate this: 1) Draw a line from (i-1)P and (i)P (For just P find the intersection from tangent line, as long I remember) 2) Find the intersection of this 2 points, called Q 3) Find the opposite of this point on the curve F(just draw a line with 90 degrees angle, and find the intersection) 4) Show this point and called it (i)P i is just number from 1 to some k

Glad you liked!!! 😎👌🏻

@@SobTim-eu3xu yes, we actually included these calculations in our first draft of the script, but then we noticed that it would be too long… and so we had to prioritize other things

@@dibeos yea, I agree with you, but showed you how to do that for part 3 maybe one day)

*q must be odd. Can be used to check Mersenne primes

Nice, I’d say that a good exercise (that may help you as well) is to try to prove Fermat’s Last Theorem for n=3, since there are some important concepts in the proof that are heavily used in Cryptography, for example.

😂😂😂 nice to see that the video is being useful!

Thanks Riley, yeah it is indeed interesting how math (especially the study of prime numbers, which are the building blocks of all other numbers) can have such amazing applications. I’m pretty sure there are many more interesting applications out there waiting for our creativity to catch up to the opportunities provided by this awesome field

Math is the language of cryptography

@@SobTim-eu3xu yeah, and I think many people do not understand that. People think that it is just a “computer algorithm”, but they do not understand how much number theory and logic is involved in creating the main cryptographic methods

@@dibeos Yes, and in some cases is not only cryptography and number theory, but maybe all the math Set Theory, Probability Theory, Computational Theory, Linear algebra, Math Analysis This is main, that I remember and know usage for cryptography algorithms

Are you talking about Sofia interacting with me (Luca)? Haha

@@dibeos yes

Aren’t they married?

@duckymomo7935 yes, we are 😎

Thanks Julian 😎🤙🏻 let us know what else you want to see in the channel. We were thinking about publishing a video about Category Theory since a few people wrote in the comments that they are interested in this field. Or would you prefer us to dive deeper into number theory and cryptography?

​​@@dibeoscategory theory is great and i would very much enjoy watching your take on it. It does seem hard to motivate though, as the original motivation came from the idea of natural transformations and how to formulate and study them. (For example there is a natural transformation between the functors that turn (A,B,C) into (A×B)×C or A×(B×C) where A,B,C are sets.) What i am currently lacking is an intuition about modern geometry. How the greeks thought about geometry (wich you already covered in your topology video) seems to be so disconnected from something like algebraic geometry to the point i don't even know why algebraic geometry has geometry in its name. (Also, why the hell do you add "points at infinity" in projective geometry?)

@@julianbruns7459 yeah, adding points in infinity is a way of extending the addition of two “normal” points, but it is not intuitive at all. I think we will make a video on Algebraic Geometry, showing how much of it is Algebraic and how much is Geometry, and why

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 😉👌🏻

Thanks! Please tell us what content you’d like us to post 😎

@@dibeos can u make a machine to change time? Like one to filter years and years of chaotic data down to a constant live feed of proofs in maths with the stories of the world put in to help with personal analogies to help get the point across of how it all might work ?

@@codybarton2090 can you explain better the idea, please? We might do it…

@@dibeos u know the steel ball plinko experiment and the laser split experimental as well as newtons refraction of light can all be used to describe chaotic spread of information over time but in a world of no resistance some things bounce back and some get stuck how would follow that flow of information even if it was correct or not it still might be valuable in time

Like a lot of apps and websites and live feeds connected to ur phone connected to WiFi Bluetooth etc but connected to an outside computer to filter and grab the information before it gets stuck or bounces back or “lost” in the hypothetical sense

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.

Really? Why do you say so? Can you expand on that (I’m curious now)?

Got it! Thanks, in the next videos I’ll say orthoNORmal 😉👌🏻

Sorry, I did not understand, but I’m genuinely curious. Can you explain?

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! 😎