Great stuff, first time learning orders here, and it is definitely a little confusing to grasp. I will however take it slow and steady. Thank you for the content !
I wonder why you didn't just show that 2^1 = 2, 2^2 = 4, 2^3 = 8, 2^4 = 3, 2^5 = 6, 2^6 = 12, 2^7 = 11, etc and show that 2 really does have an order of 12. Also, 2^12 = 1 (mod 13) but when you did your deduction you wanted to find out if there was no 1 in your list. I can understand your technique it is interesting. I think once you saw 2^6 = 12 you knew that the order had to be 12
Wait never mind, I think its because to find a primitive root, we are solving the equation r^phi(n) = 1(mod n), this is the exact statement of eulers theorem, and hence we must have gcd(r,n) = 1.
When showing these examples, you should take time to actually explain the logic and reasoning for why these concepts exist. What's the purpose of them? Why are we doing all these arrangements with numbers? What is our GOAL with primitive roots?
@@tomatrix7525 Its not that he is struggling, he is trying to find a practical application as to why we need to know this. Truth be told I don't see why this is practically useful. A large branch of mathematics doesn't have many applications, other than fun and glory !
No, because 1^1 ≡ 1 (mod n). In other words the order of 1 (mod n) is always 1, recalling that the order of a (mod n) is the _smallest_ power of a that is congruent to 1 (mod n) . For a primitive root, r, by definition we have ord_n(r) = φ(n), and φ(n) > 1 for all n > 2. That means the only natural numbers with a primitive root of 1 are 1 and 2.
Can someone explain me why 1 is excluded after the lecturer said that we should take a number and check if exponent of that number gives 1 (mod n)? Does 1 not satisfy this? ruclips.net/video/aW0b_lCzOCQ/видео.html
It looks like the goal at ruclips.net/video/aW0b_lCzOCQ/видео.html is to find the primitive roots. 1 cannot be a primitive root because its ord_n = 1 which is smaller than phi(n).
Wish RUclips provided a way to limit search to specific countries. So difficult to find proper English-speaking computer science videos, due to the enormous number of crappy Indian videos.
Bro dropped the info I needed to understand the content then just went peace and dipped lol
Great stuff, first time learning orders here, and it is definitely a little confusing to grasp. I will however take it slow and steady. Thank you for the content !
Wow that's really an easiest explanation ❤❤❤❤❤
Part of me wants to sleep, part of me wants to go read a romance fanfic... but here I am telling them both "NOT YET!"
Professor Penn, thank you for a great introduction to Primitive Root Modulo N. I will review this video for a deep understanding of this subject.
Your content is great as always!
I wonder why you didn't just show that 2^1 = 2, 2^2 = 4, 2^3 = 8, 2^4 = 3, 2^5 = 6, 2^6 = 12, 2^7 = 11, etc and show that 2 really does have an order of 12. Also, 2^12 = 1 (mod 13) but when you did your deduction you wanted to find out if there was no 1 in your list. I can understand your technique it is interesting. I think once you saw 2^6 = 12 you knew that the order had to be 12
Why you said there is no primitive roots in last problem
Because everything has order 2 modulo 8, but to be primitive root we would need an integer of order 4 modulo 8.
And (Z/8Z, ×) is precisely a Klein Four Group, whose elements are self-inverse.
Using too much mathematical "slang" to explain a concept that is simpler than the actual language utilized...
Why do we have that the possible primitive roots must be relatively prime to our modulus?
Wait never mind, I think its because to find a primitive root, we are solving the equation r^phi(n) = 1(mod n), this is the exact statement of eulers theorem, and hence we must have gcd(r,n) = 1.
When showing these examples, you should take time to actually explain the logic and reasoning for why these concepts exist. What's the purpose of them? Why are we doing all these arrangements with numbers? What is our GOAL with primitive roots?
Dont bother. Its inteligent - idiot type, just watch his moves...
It’s easy. If you’re struggling here you should go back to the basics of euler totient functions and modular stuff
@@tomatrix7525 Its not that he is struggling, he is trying to find a practical application as to why we need to know this.
Truth be told I don't see why this is practically useful. A large branch of mathematics doesn't have many applications, other than fun and glory !
@@prathikkannan3324 don’t study pure math if you want practical applications outside of computer science.
Thank you so much!
Thank you so much.
Why does the order of an integer always divide eulers totient of it?
Shouldn't 1 be a primitive root any (mod n), because 1^phi(n) = 1(mod n ) for all phi(n) ?
No, because 1^1 ≡ 1 (mod n). In other words the order of 1 (mod n) is always 1, recalling that the order of a (mod n) is the _smallest_ power of a that is congruent to 1 (mod n) .
For a primitive root, r, by definition we have ord_n(r) = φ(n), and φ(n) > 1 for all n > 2. That means the only natural numbers with a primitive root of 1 are 1 and 2.
one word: best!
Sir pls change ur background picture..... It change our concentration
Can someone explain me why 1 is excluded after the lecturer said that we should take a number and check if exponent of that number gives 1 (mod n)? Does 1 not satisfy this?
ruclips.net/video/aW0b_lCzOCQ/видео.html
It looks like the goal at ruclips.net/video/aW0b_lCzOCQ/видео.html is to find the primitive roots. 1 cannot be a primitive root because its ord_n = 1 which is smaller than phi(n).
Thank you!!
wait what's a mod? Shit, just when I think I'm smart I gotta go back and learn something else new first. haha, that's okay, I'll find it, thanks!
Wish RUclips provided a way to limit search to specific countries. So difficult to find proper English-speaking computer science videos, due to the enormous number of crappy Indian videos.