Number Theory is Impossible Without These 7 Things

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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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[hiu]

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!

[khaledqaraman]

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 📏

[KAn-brpy]

man proceeds to show a picture of Japan when he says China. I mean i love ur video but come on man X_X

[OpPhilo]

Thank you very much sir and mam.
Love from India.

Your videos very helpful for me. I learn more things through your videos

[talastra]

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.

[alextrebek]

I appreciate your videos, thank you! Would you ever be open to covering semi-rings (tropical or otherwise), or Category Theory?

[purplepenguin]

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.

[rileythesword]

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!

[jonathanlister]

Another "Stoater" as we say in Scotland.

[swingyflingex]

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.

[buckleysangel]

The square root of phi squared plus one squared equals phi squared. Every integer version of this is similar to the Fibonacci sequence. ❤

[shaneri]

Thank you Luca and Sofia! You are great!!

[omarserranososa]

Just subscribed. Please keep making these videos. You all deserve much more views. This content is gold

[SobTim-euxu]

As mathematician in Number Theory I love this video🥰
Also I happy for mentioning Cryptography, bc I cryptographer also😍

[hassankhamis]

• Pythagorean
• Remainders
• Primes
What else ?

[KeithKessler]

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.

[williejohnson]

: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.

[tobyendy]

Great video, could we have one talking about thermodynamics or complex number

[jbangz]

Andrew Wiles' persistence is unbelievable, I hope I could tackle one of the major unsolved mathematical problem.

[lunarthicclipse]

Love thr vid! Very interesting topic and I learned quite alot! You deserve more subs❤