The intuitive reason is that the numbers before being squared are either multiples of 3 or missing being multiples of it by 1 (either by -1 or +1). So the squares are always gonna have the form 3k when they are already multiples or they'll be 3*a+-r which (r is at most 1, so...) when squared gives 9a²+-6ar+r². Since r² = 1 at most, then we can always write 3(3a²+-2ar)+1 with 3a²+-2ar=k which is the proof, basically, as given in the vid, with less fuzz. If you think about it, it's actually trivial.... Why I never noticed I don't know.
Hello do you know of any books that teach basic proof knowledge (or discrete mathematics) along with precalculus/algebra and trigonometry? I want to learn proofs, but I don't want to invest in learning a whole new proof book. So if there was a book like that, it would be like two birds with one stone. Another advantage with a proof plus precalc book is that I leave out with a better understanding of those topics as the proof work would force me to internalize those concepts and know more about the 'why' behind the math.
Hmmm not really no:) I feel like most of the proof books just focus on the structure of proof writing. They start with logic, then talk about sets, then give several sample proofs in various areas of math. A lot of times those example proofs do involve algebra, but I really don't think a book like that would make you better at algebra. I think for basic algebra/precalc skills it's just better to get some books on those topics and do practice problems.
th-cam.com/play/PLHXZ9OQGMqxersk8fUxiUMSIx0DBqsKZS.html this can be a good start if you want to learn proofs. i wouldnt recommend starting with calculus or sth like that without a good foundation.
come to your own conclusions about the "why" but all you really need is Proofs by Jay Cummings and do the exercises -- you will see the rules and logic that underpins basic precalculus rules and properties.
Interesting, I have never heard of the division algorithm. I would have used the relative prime algorithm. The proof is much the same. Stating all integers are of the form a = b or b + 1 or b + 2 === 3c & 3c +/- 1... so 3k = a^2 or a^2 - 1 or a^2 + 1 ...
For most colleges the sequence is College Algebra, Pre Calculus, Calculus 1, Calculus 2, Calculus 3, Linear Algebra, Differential Equations. A student who has never taken Pre Calculus but plans on taking differential equation inorder to obtain his bachelors. If the student wants to jump straight into higher level math courses without ever taking Calc 1,2,3 or Pre calc, or even College Algebra. The purpose for this is because I want to self study and do a pre req exam so I can be done with my math classes quicker. Obviously it would be ridiculous but would it be possible for the student if they were to self study? Is it even possible to self study up to those higher math levels? What can the student do to maximize their chances to be successful with the process? Thank you! I recently discovered your channel, very optimistic but also straight forward.
It’s definitely possible. If you aren’t familiar with precalc then you’ve got a long way to go, but you don’t have to take university classes to acquire that knowledge. The advantages that classes provide are accreditation and structure. There are many resources online like Khan Academy, Brilliant, TH-cam. Also your local library or pdfdrive for textbooks. Textbooks are the best because they contain a ton of problems, and you’re gonna have to do a lot problems
It depends on your university and your ability to self study It is absolutely possible to self study pre-calc, calc 1-3, linear algebra, and diff eq. It is possible to self-study any math class if you are dedicated enough. If you think about it, math research is basically people studying math we aren't even sure exists just yet and they form new theorems and lemmas that are used to continuously make math bigger, so if someone can technically study math that doesn't exist yet, then people can study any math class that is well defined for sure College algebra is normally a pre-req for pre-calc which is a pre-req for the calculus sequence. Calc 2 at many schools is a pre-req for linear and diff eq, however, you can do linear algebra with very little calc knowledge. If you finished pre-calc, know what a limit and derivative is, that is probably enough to read most intro linear algebra books. I'd say at least calc 2 is required for diff eq, but I['d recommend also having both calc 3 and linear algebra under your belt because it will make diff eq easier, but they aren't required for intro diff eq classes (just know what a partial derivative is which is basically the start of calc 3) If your university lets you test out of all these classes, that is great! If you have the drive you can definitely self study and get through. However, personally, I'd rather take the classes. It is very helpful to have a professor to go to if you are struggling, especially if you have a good professor. My college only lets you test out of everything below calc 1 (without AP credit from highschool) so the highest you can place into without AP/IB credit is calc 1. If your school is different, by all means feel free to try!
I think that instead of writing n=3k+r for r=0,1,2 it would have been better to write n=3k+r for r=-1,0,+1. This unifies the proof in the case of r=-1 or +1.
doing proofs is when I totally lost interest in math -- on the hand, the math exam you showed from the US Navy book last night is the kind of stuff that gets my interest going high -- that's the kind of stuff that's interesting -- sadly, I know that kind of math wouldn't be possible without proofs :(
was looking for a pattern in dropping all trivial cases of N square = 3k+0 thinking in distance from previous k 1^2 1 = 3*0+1 ---0 2 -- 4 = 3*1+1 ---1 4 -- 16 = 3*5+1 ---4 5 -- 25 = 3*8+1 ---3 7 -- 49 = 3*16+1 ---8 8 -- 64 = 3*21+1 ---5 10 -- 100 = 3*33+1 ---12 11 -- 121 = 3*40+1 ---7 13 -- 169 = 3*56+1 ---16 14 -- 196 = 3*65+1 ---9 16 -- 256 = 3*85+1 ---20 17 -- 289 = 3*96+1 ---11 19 -- 361 = 3*120+1 ---24 20 -- 400 = 3*133+1 ---13 enough -- we got the gist but it revealed something I didn't know kinda cool -- for all primes > 3 primes squared minus 1 are 24X if you notice > 3 primes squared land on 3*8K+1 not only that, if it is prime squared , it will not be found anywhere but on 3*8K+1 using maplesoft we can see the trend continues, for example: 99999999999999999989 is prime squaring equals 9999999999999999997800000000000000000121 minus 1 / 24 is 416666666666666666575000000000000000005 an integer 3030303030303030303030303030303030303030303030302996633 is a prime (3030303030303030303030303030303030303030303030302996633^2 - 1)/24 382614018977655341291704928068564432200795837159465020576981940618304254667891031527395163758800122483722362 an integer this is not for say a primality tester and other values squared -1 are 24X as well such as 49^2 - 1 = 2400 but I thought it interesting for all primes
The intuitive reason is that the numbers before being squared are either multiples of 3 or missing being multiples of it by 1 (either by -1 or +1). So the squares are always gonna have the form 3k when they are already multiples or they'll be 3*a+-r which (r is at most 1, so...) when squared gives 9a²+-6ar+r². Since r² = 1 at most, then we can always write 3(3a²+-2ar)+1 with 3a²+-2ar=k which is the proof, basically, as given in the vid, with less fuzz.
If you think about it, it's actually trivial.... Why I never noticed I don't know.
Great explanation. The rearranging of the variables in the context of rigorous mathematical proofs made it fuzzy
So relaxing watching someone do a clearly written proof
I really liked this video. I look forward to seeing you do more proofs.
Thank you!
Hello do you know of any books that teach basic proof knowledge (or discrete mathematics) along with precalculus/algebra and trigonometry? I want to learn proofs, but I don't want to invest in learning a whole new proof book. So if there was a book like that, it would be like two birds with one stone. Another advantage with a proof plus precalc book is that I leave out with a better understanding of those topics as the proof work would force me to internalize those concepts and know more about the 'why' behind the math.
Hmmm not really no:) I feel like most of the proof books just focus on the structure of proof writing. They start with logic, then talk about sets, then give several sample proofs in various areas of math. A lot of times those example proofs do involve algebra, but I really don't think a book like that would make you better at algebra. I think for basic algebra/precalc skills it's just better to get some books on those topics and do practice problems.
th-cam.com/play/PLHXZ9OQGMqxersk8fUxiUMSIx0DBqsKZS.html this can be a good start if you want to learn proofs. i wouldnt recommend starting with calculus or sth like that without a good foundation.
come to your own conclusions about the "why" but all you really need is Proofs by Jay Cummings and do the exercises -- you will see the rules and logic that underpins basic precalculus rules and properties.
I proved this differently using "Calculus of Finite Differences", i.e., expressing perfect squares as arithmetic sequences.
Hello dudes. Spontaneously this comes up to me: Let x be an integer. Then x= 0,1,2 (mod 3). Thus x^2 = 0,1 (mod 3) proving this statement.
Math like this = academic music theory
Math that sends men to the moon = Charlie Parker.
Interesting, I have never heard of the division algorithm. I would have used the relative prime algorithm. The proof is much the same. Stating all integers are of the form a = b or b + 1 or b + 2 === 3c & 3c +/- 1...
so 3k = a^2 or a^2 - 1 or a^2 + 1 ...
1. For which positive integers K are the numbers K(K+1)/2
and K − 1 both perfect? Recall that a
number is perfect if it is a positive inte
As stated in the video it does not work for 1 as there is no way to write 1 whose square is also 1 as 3k or 3k+1 with k being an integer.
For most colleges the sequence is College Algebra, Pre Calculus, Calculus 1, Calculus 2, Calculus 3, Linear Algebra, Differential Equations. A student who has never taken Pre Calculus but plans on taking differential equation inorder to obtain his bachelors. If the student wants to jump straight into higher level math courses without ever taking Calc 1,2,3 or Pre calc, or even College Algebra. The purpose for this is because I want to self study and do a pre req exam so I can be done with my math classes quicker.
Obviously it would be ridiculous but would it be possible for the student if they were to self study? Is it even possible to self study up to those higher math levels? What can the student do to maximize their chances to be successful with the process?
Thank you! I recently discovered your channel, very optimistic but also straight forward.
It’s definitely possible. If you aren’t familiar with precalc then you’ve got a long way to go, but you don’t have to take university classes to acquire that knowledge. The advantages that classes provide are accreditation and structure. There are many resources online like Khan Academy, Brilliant, TH-cam. Also your local library or pdfdrive for textbooks. Textbooks are the best because they contain a ton of problems, and you’re gonna have to do a lot problems
It depends on your university and your ability to self study
It is absolutely possible to self study pre-calc, calc 1-3, linear algebra, and diff eq. It is possible to self-study any math class if you are dedicated enough. If you think about it, math research is basically people studying math we aren't even sure exists just yet and they form new theorems and lemmas that are used to continuously make math bigger, so if someone can technically study math that doesn't exist yet, then people can study any math class that is well defined for sure
College algebra is normally a pre-req for pre-calc which is a pre-req for the calculus sequence. Calc 2 at many schools is a pre-req for linear and diff eq, however, you can do linear algebra with very little calc knowledge. If you finished pre-calc, know what a limit and derivative is, that is probably enough to read most intro linear algebra books. I'd say at least calc 2 is required for diff eq, but I['d recommend also having both calc 3 and linear algebra under your belt because it will make diff eq easier, but they aren't required for intro diff eq classes (just know what a partial derivative is which is basically the start of calc 3)
If your university lets you test out of all these classes, that is great! If you have the drive you can definitely self study and get through. However, personally, I'd rather take the classes. It is very helpful to have a professor to go to if you are struggling, especially if you have a good professor. My college only lets you test out of everything below calc 1 (without AP credit from highschool) so the highest you can place into without AP/IB credit is calc 1. If your school is different, by all means feel free to try!
Fun fact: Every square of an integer is also of the form 2k or 2k+1 :v)
🤡is that even a fact 🤡 I just came here to know why any square is not of the form 3k+2 or 3k-1
now I can sleep peacefully , thank you
I think that instead of writing n=3k+r for r=0,1,2 it would have been better to write n=3k+r for r=-1,0,+1. This unifies the proof in the case of r=-1 or +1.
doing proofs is when I totally lost interest in math -- on the hand, the math exam you showed from the US Navy book last night is the kind of stuff that gets my interest going high -- that's the kind of stuff that's interesting -- sadly, I know that kind of math wouldn't be possible without proofs :(
was looking for a pattern
in dropping all trivial cases of N square = 3k+0
thinking in distance from previous k
1^2 1 = 3*0+1 ---0
2 -- 4 = 3*1+1 ---1
4 -- 16 = 3*5+1 ---4
5 -- 25 = 3*8+1 ---3
7 -- 49 = 3*16+1 ---8
8 -- 64 = 3*21+1 ---5
10 -- 100 = 3*33+1 ---12
11 -- 121 = 3*40+1 ---7
13 -- 169 = 3*56+1 ---16
14 -- 196 = 3*65+1 ---9
16 -- 256 = 3*85+1 ---20
17 -- 289 = 3*96+1 ---11
19 -- 361 = 3*120+1 ---24
20 -- 400 = 3*133+1 ---13
enough -- we got the gist
but it revealed something I didn't know
kinda cool -- for all primes > 3
primes squared minus 1 are 24X
if you notice > 3 primes squared land on 3*8K+1
not only that, if it is prime squared , it will not be found anywhere
but on 3*8K+1
using maplesoft we can see the trend continues, for example:
99999999999999999989 is prime
squaring equals 9999999999999999997800000000000000000121 minus 1 / 24 is
416666666666666666575000000000000000005 an integer
3030303030303030303030303030303030303030303030302996633 is a prime
(3030303030303030303030303030303030303030303030302996633^2 - 1)/24
382614018977655341291704928068564432200795837159465020576981940618304254667891031527395163758800122483722362
an integer
this is not for say a primality tester
and other values squared -1 are 24X as well such as 49^2 - 1 = 2400
but I thought it interesting for all primes
Thank you very much. May God Bless you
Is the Division Algorithm also called Euclid's Division Lemma?
Could you do this one?
Prove p^3 + (p-1)/2 cannot be a perfect square for p>2 and p is prime
What about if i let b=othr integers?
Did you ever read Calculus by G Tewani? . Its very nice book 📕
Thank you bro... Iam a malayali ...❤
Proof Ramanujan mock theta function
Lol i just did this problem 2 days ago, in the introduction to number theiry
Awesome !
Superb👏
जय श्रीराम