Solving a Mod Equation with Prime Powers

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20 апр 2021

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Комментарии : 71

@SyberMath 3 года назад

Basics of Modular Arithmetic: ru-vid.com/video/%D0%B2%D0%B8%D0%B4%D0%B5%D0%BE-Q_V_itu_kbs.html

@matchedimpedance 3 года назад

In hindsight, x^2 + x + 1 = 49 (x-11) mod 49. So x^2 - 48x + 540 = 0 mod 49 So (x-18)(x-30) = 0 mod 49. So x is either 18 or 30 mod 49. But that is hard to see in advance!

@SyberMath 3 года назад

That's pretty good! I agree. This is hard to see. Maybe one approach might be to turn x^2 + x + 1 into x^2 - 48x + 1 + 49k and then look for an appropriate k value to make it factorable. The discriminant, 2300-196k, needs to be a perfect square which means k=11

@matchedimpedance 3 года назад

@@SyberMath Careful. I think the discriminant is 2300-196k. That is a square when k = 11.

@SyberMath 3 года назад

@@matchedimpedance Yess! 😁😎

@davidgillies620 3 года назад

The sets of roots of polynomials modulo a prime power is a surprisingly deep topic in number theory. A non-obvious result is that for n = p^k ≠ 4, p prime, not all subsets of ℤ/nℤ can be roots of polynomials in ℤ modulo n. In other words, there exists a set of integers modulo n such that you cannot construct a polynomial of _any_ finite degree with integer coefficients whose roots modulo n are those integers.

@242tommy3 3 года назад

you are very skillful at this, great job solving this problem

@atharvadinkar8847 3 года назад

Amazing (as always)!! Just wanted to point out one cool thing that I noticed: When studying Quadratic equations with real numbers, we're taught that if the QE is of the form x^2+ax+b = 0, then the sum of the roots is (-a) and the product of the roots is b. It should come as no surprise that this holds true here as well - the roots were 18mod(49) and 30mod(49), which add up to give (-1)mod(49) and multiply to give 1mod(49). Can you please do more videos on number theory, like this one? Additionally, can you shed some light on Galois Fields? I'm not sure how exactly they work, but I know that they have something to do with how multiplicative inverse are defined in modular arithmetic.

@tonyhaddad1394 3 года назад

I think its a coinsidence for vietas formula to happen here beacaus its another concept (when the sum of the root equal the coeffition and the product equal the last coeffition that is the vieta formula ) if im wrong tel me !!!!

@atharvadinkar8847 3 года назад

@@tonyhaddad1394 I don't think it's a coincidence. Let the roots be x1 and x2. The QE would be (x-x1)(x-x2) = 0mod(49). Simplifying, we get that the middle coefficient is the negative of the sum of the roots, mod49 and the last coefficient is the product of the roots mod49.

@manojsurya1005 3 года назад

I never seen a quadratic in a modulo problem,everyday u give us a good variety of problems 😃,keep doing this.

@SyberMath 3 года назад

Thank you, I will

@aashsyed1277 3 года назад

Good video

@242math 3 года назад

love the way you tackle these complicated problems, here to watch and learn

@deepjyoti5610 3 года назад

Nyccc problem and elegant explanation

@mohamedihab4599 3 года назад

Great problem! Also your way of explaining is easy and clear, thank you very much. If I could ask a question, how long in your opinion does it take to be quite good at number theory?

@SyberMath 3 года назад

Depends. I took Number Theory after I graduated college. Take a class from a good professor and find some good textbooks and start studying. It should probably take you a few months again depending on how much background you have and how much time you put out there.

@larzcaetano 3 года назад

Amazing video!! Do you have plans on deriving some formulas in your channel? I would love to see your approach and derivations are not a common topic in RU-vid.

@SyberMath 3 года назад

Why not?

@akshatjangra4167 3 года назад

Take that math elite!

@GustavoCarvalho-qv3od Год назад

what do i need to do if there's a variable "x" in the left and right side of the equation? how do i move it?

@carloshuertas4734 3 года назад

Another great mathematical problem!

@elifalk8544 Год назад

Once you have one solution, use the fact that they add up to -1 (based on the details of the equation, this works for regular numbers too) which in mod 49 is equal to 48. The second answer is extremely simple after that.

@SyberMath Год назад

Right

@tonyhaddad1394 3 года назад

Amazing !!!! Good job

@SyberMath 3 года назад

Thank you so much 😀

@aashsyed1277 3 года назад

I love this

@asmocak1053 3 года назад

You can use hensel's lemma for this problem.

@SyberMath 3 года назад

Great! Thank you! I was waiting for this comment!!! 🤩

@ramaprasadghosh717 2 года назад

very nice approach

@SyberMath 2 года назад

Glad you think so! 💖

@chessdev5320 3 года назад

please factorise (u⁵+u+1)

@Happy_Abe 3 года назад

What about for other values of k and n that would satisfy the equations? Don’t they give other values for x?

@SyberMath 3 года назад

No they don't

@Happy_Abe 3 года назад

@@SyberMath how would one prove they’re the same regardless of the choice of k

@binamahadani3267 3 года назад

n=2 gives a satisfying solutions

@binamahadani3267 3 года назад

Thx

@carloshuertas4734 3 года назад

Explain it well.

@DolanWay Год назад

If the mod is not 0, but a positive integer such as 1, will the method work?

@SyberMath Год назад

I think the mod has to be 2 at least

@miro.s Год назад

There are 4 solutions, not only two!

@samiulfahim5384 3 года назад

Splendid 👍

@SyberMath 3 года назад

Thanks ✌️

@MathElite 3 года назад

Second! Kudos to Akshat Jangra for beating me Nice video again! You always create great stuff

@SyberMath 3 года назад

Thanks again!

@aashsyed1277 3 года назад

Do you love maths? Math elite?

@aashsyed1277 3 года назад

Math elite how old are you?

@MathElite 3 года назад

@@aashsyed1277 yes i am math elite love is a strong word....just kidding

@MathElite 3 года назад

@@aashsyed1277 19 years young

@damiennortier8942 2 года назад

Why not solve the problem as x^2 + x + 1 = 0 and add 49 to the solution?

@miro.s Год назад

Not possible because of the linear shift given by constant 1.

@binamahadani3267 3 года назад

Let X= 6,then x^2+X+1=43

@stsaxa584 3 года назад

You could also prove that there aren't any other solutions.

@SyberMath 3 года назад

How?

@asmocak1053 3 года назад

@@SyberMath lagrange already proved that 😁

@stsaxa584 3 года назад

@@SyberMath prove that if x-18=0 mod 7 then x-30 !=0 mod 7 and if x-30=0 mod 7 the x-18 !=0 mod 7. Then conclude that (x-18)(x-30)= 0 mod 49 implies x-18=0 mod 49 or x-30=0 mod 49.

@miro.s Год назад

There are two other solutions!