Divisibility Mathematical Induction Proof: 3 Divides 2^(2n) - 1

In this video I do an induction proof with divisibility. I prove that 3 divides 2^(2n) - 1 for all positive integers n.

I hope this video helps someone:)

[magnusdc]

I'm a total math-illiterate trying to learn discrete mathematics. And these videos of yours makes so much more sense to me than a lot of the learning material that i try to dig into. Thank you for taking the time!

[TheAAZSD]

Watching your videos keeps the blade of proofs sharp. This channel isn't nearly as watched as it should be.

[quitecomplex]

It’s a nice proof and a great use of proof by induction, but I don’t think induction is necessary to prove the result. A much simpler proof is that since x^2-1=(x-1)(x+1), 2^(2n)-1=(2^n+1)(2^n-1) Now upon division by 3, 2^n cannot have a remainder of 0 (for it has no prime factor 3) it must either have a remainder of 1 or 2. Suppose it has a remainder of 1. Then 2^n-1 is divisible by 3. Likewise, if it has a remainder of 2, then 2^n+1 is divisible by 3. Hence 2^(2n)-1 must be divisible by 3.

[mlfacts]

Great stuff! Absolutely enjoyed watching you proving it . When I first was taught PMI, I didn't like it and had a really hard time with it . But now, I have a much better understanding of it . Thank you very much the Math Sorcerer .

[tsshamoo]

Proof w/o Induction

Let q=2^n. Then

Now consider the numbers q-1, q, and q+1. These are 3 consecutive integers, so one of them must be a multiple of 3. But we know that number cant be q, since q=2^n. Therefore 3 divides either q-1 or q+1 which means 3 definitely divides (q-1)*(q+1) and were done

[pinklady]

Few weeks ago, I had been doing exercises of arithmetic & geometric series. Now mathematical induction today. Great.

[nyankiku]

Yooo this exact question was in my review session today and I'm 100% sure its going to be on my exam tomorrow. Thank you dear teacher, for you have shown me the way!

[timhounsgaard]

This helped me understand induction with divides thank you so much

[valeara]

Genial, locamente, con los videos explicativos en español sentí que no entendía nada :(
Solamente habían ejemplos muy difíciles.
Fuiste muy amable para explicar <3 tkm

[justinclark]

Just a reminder that I ONLY like math because of my luck in having you as a teacher.

[mobilelegendsstudio]

God bless you teacher, your question came out in my online exam.. Thank you very much

[firelordify]

Thank you! I was struggling with PMI, and you made it click.

[motiversity_]

idk how much I can thank you
thank you so much

[sasiprakash]

Thank you so much man.You are really awesome.

[Finne]

Very cool induction proof that I understood for once! Great hair!

[simons]

We can also do just modulo 3 and get: 4^k-1==1^k-1==0 (mod 3)

[georgettebeulah]

A multiple but it look so real I am a bit late i can understand so much from this sense

[jaanuxlicious]

Well, now I know why it is like x+3x. I feel so dumb 😂. Thank you