A general strategy for solving Pell's equations (ft.

[@blackpenredpen]

What are the tens and the units digits of 7^7^7? Solution kzbin.info/www/bejne/lYCkd6yBj7yqr5osi=npITrvCxthrMc_QN

[@amansparekh]

What this links to is units in the ring Z[rootk]. Every unit has norm 1 and thus is a solution to the equation. Clearly units form a multiplicative group.

[@paultijink4235]

Great! Please do make that video showing there are no other solutions!

[@randomjin9392]

But now the most important part: how do you find that "first solution" - and more importantly, enure that it is the first (i.e. smallest)? Guess and check sounds very involved for equations with non-trivial coefficients, e.g. x²-73y² = 1217

[@acelm8437]

Exactly, how do you even know a solution exists for all nonsquare k?

[@SyberMath]

You can use continued fractions to find the smallest solution and then build the others from that

[@VideoFusco]

​@@SyberMathi don't it is so obvious how to use continued fractioj for that

[@ssgamer5693]

That notebook was the least asian thing in the video.I can confirm that every competitive exam aspirant prefers longbooks(400pgs+) for exam prep because it is easier to maintain and search old things.

[@ComrdFinisher-zb4kq]

Nice to meet you ❤ Tomorrow morning I have mathematics exam wish me success

[@SoBasicallyImRetarded]

Same here

[@cdkw2]

good luck!

[@alphazero339]

Good luck! Calc exam?

[@ComrdFinisher-zb4kq]

@ of course bro ,thank you

[@triniasta]

good luck,,

[@Blaqjaqshellaq]

Squaring 9+4*5^(1/2) gives the pair {161, 72}. (I was waiting for you to give that pair, but I had to figure it out myself.) You can also use this method to give rational approximations of the square root of 5: 9/4, then 160/72=20/9...

[@thesupergamergr3255]

Happy new year y'all!! 🎇🎇🎇🎆🎆🎆

[@Happy_Abe]

Wouldn’t negating x values also give solutions so we also have all the reflected values

[@aggelosaggelos8243]

so what? this is quite obvious since x^2=(-x)^2 for all reals. we are looking for the solution where x and y have the smallest absolute values they could have and we take the positive one for simplicity's sake. we say that the first solution (x1,y1) is the one for which |x1|

[@Happy_Abe]

@ yeah I’m just saying these are solutions too that weren’t mentioned since it was asked if these are all the solutions in the video, wasn’t a major point.

[@devrimturker]

Log Ladders, Lemmermeyer's Product

[@Budgeman83030]

I may not understand the math completely but the video is amazing to watch

[@asifulalamc]

Amazing video! Happy new year to everyone. 🎉🎆

[@hamdamoverali]

Very well explained 😊

[@johns.8246]

OK, now how about doing this for x^2 - 61y^2 = 1 ?

[@GreenMeansGOF]

I want to see the proof! Also, can this be extended to when the right hand side is composite?

[@pzelact4328]

1:32 later is when? give me timecode so i can watch and understand every step not just seing something and forgeting about it

[@johns.8246]

And for that matter, is there a simple way of finding all integer solutions for elliptic curves of the form y^2 = x^3 - x + r^2 ?

[@Nikkikkikkiz]

In my classes, we solve this by having the solution be a continued fraction and just write answer in continued fraction format

[@scottleung9587]

Nice proof!

[@jakobthomsen1595]

Very interesting!

[@khoozu7802]

The pell equation x^2-2y^2=+-1 gives approximation of square root 2

[@OubleJum]

I dont understand anything, I just watch these to fall asleep

[@kornelviktor6985]

The thumbnail is so cool

[@SDCORL]

Secant ,tangent/k are all of the solutions.

[@Choco_Black69]

張耕宇學長好！

[@octopuskeng]

你好⊙ω⊙😊

[@elecbertelecbert-l5e]

The Chinese are very smart thank you

[@trelligan42]

Sorry, I really need comments for this gentleman.

[@kimsmoke17]

Can you find a room with more echo? 😂

[@JikoAmonus]

im in 9th grade, I cant even do basic calculus

[@محمدبورايب]

it's not important for you, don't do it

[@prabhakarsingh6821]

"Do not worry about your difficulties in Mathematics; I can assure you mine are still greater" - Einsteine

[@محمدبورايب]

@@prabhakarsingh6821 thank you i will use this quote later every there and where

[@JollyBooger]

🎉🎉🎉

[@sergio4425-g7u]

존나잘생겼네

[@Maths786]

If "f" is a differentiable function on the interval [0, 1], with the following boundary conditions: f(0) = 0 f(1) = 1 Then, find the minimum value of the integral: ∫ from 0 to 1 of (f'(x))² dx.

[@debtanaysarkar9744]

1

[@fuzzylostallrightstoexist]

Zamn

[@epicstar86]

peak

[@abdulrahmanabukharma7969]

kzbin.info/www/bejne/q6LadGdnh6mDr8k

[@nouarislimani]

X=i^4 is solution of sqrt(x)=-1

[@KewlWIS]

its not

[@nouarislimani]

Sqrt(i^4)=i^2=-1

[@KewlWIS]

@nouarislimani sqrt(i^4) = sqrt(1) = 1, youre thing is wrong cuz whenever you do sqrt(x^2) its always equal to |x|

[@markerguy]

In mathematics we have a definition- √(x)² = |x|. Here it's √i⁴ = √(i²)² = √(-1)² = |-1| = 1. So this isn't possible.

[@markerguy]

Also with that logic in mind you can say that (-1)² is a solution too.

[@yukfaicheung7484]

I will find god

[@gobindaprasadpoudel]

He has good knowledge of mathematics I had never learned till 10

[@cy3er_hawkzz287]

im too fast

[@adamcionoob3912]

wow, that's so cool

[@SudhakarSudhakardontineni]

Happy New Year🎉❤ 🎂🎊💐👑💵💵💵💵💵💵🧠🧠🧠🧠🧠🧠📐(^o^) to MM is MATHEMATICS MASTER