The Most Misleading Patterns in Mathematics | This is Why We Need Proofs 

Hey guys, at 4 minutes in I forgot to put limits on the integral. It's supposed to be the integral from 0 to infinity of those trig functions (just like that first integration we saw). My mistake!

I conjecture that all numbers are less than 10^100. I don’t have proof, but I’ve checked everything up to 10^99 and it seems to hold, so it must be true.

Well you're right we need proofs, that's why I have this really elegant proof for my theorem, however, this comment is too small to fit it in. - Fermat

Statisticians: p < 0.05 is good for me. MajorPrep: All is lost

2:31 If u travel to the top of the paper stack at the speed of light, it would take you no time at all, because your time stops! *puts sunglasses on

This is why in math you have to prove *everything* before generalizing it. And now it should be clear that theorems in math are necessary (many students think they're useless)

Wait, if it takes so long to travel along that stacked paper, then who stacked that paper in the first place?

Not often do I come across a maths channel that I like instantly. Today's one of these days

When you don't include rules of how a question is to be answered, all answers which satisfy the requirements in the question are correct. Thus, the answer to your opening question is not 31 - rather one possible solution of many is 31.

MajorPrep, I just want to thank you so much man! I started following you about a year ago ( 11th grade ) . I am from Romania and because of you I decided I wanted to major in EEE. Now I got accepted by 4 Universities in the UK. Wish you the best, you truly have inspired me to choose this career! Cheers

03:30 You forgot to say that the numbers are the MAXIMUM number of regions. From n=6 you can get fewer regions depending on their location on the disc's rim. Point in case: if you nudge the lower point in the n=6 case a bit to the right, you can make the smaller center triangle disappear, and have only 30 regions. (You also have only 30 regions if the points are evenly distributed around the rim.)

You: *picks any number* Collatz conjecture: I am inevitable

I'm a graphic designer/writer and for the most part, the furthest thing from a mathematician. However, this pissed me off beyond belief. Thanks for that!

a better and general way of representing the collatz conjecture would is "Show that picking any positive integer if you apply those rules it always reaches a number that is a power of 2 so, the number can be represnted by 2^x where x is a positive whole number." because if it reaches a power of 2 , repeatedly dividing it by 2 will eventually land you at 1 inevitably.

So thats why maths exams are filled with proof questions... interesting

_Vietnam-like flashbacks to series and sequences_

This was an EXCELLENT video! It really underscores the importance of not jumping to conclusions too soon in math. And, haha, your expression at 2:28 mirrored mine at that moment! My question is: how in the world do mathematicians discover these breaks-in-pattern at these astronomically large numbers??? Crazy!

I'm surprised Skewe's Number never made an appearance. Sometimes the first counterexample even in a pattern that arises naturally can be extremely large.

2:26 3:59 This is a bit of a cheat because the integrals continue to have the value $\pi/2$ so long as the denominators sum to less than 2. It's straightforward to show that the harmonic series 1+1/2+1/3+... diverges. It just takes more terms if you use 1+1/101+1/201+... If you're interested: Borwein sinc integrals.

Love your recent math videos! Keep up the good content!

4:21 Zach: the answer = pi/2 - 2,31*10^-11. Other engineers: no aproximation, wait that’s illegal.

Very happy I’ve discovered this channel! You explain things really well!

One of my favs is the expansion of (X-1)(X-2)(X-3)(X-4)(X-5) as you can see you get a poly to the 5th power. But if you let X=1,2,3,4 or 5 You get zero. Once you let X=6 it does not equal zero. So you cant make assumptions.

At the beginning I guessed 31 from 3Blue1Brown and then you said “exactly” I was like Ok

One of the best videos on Mathematics I've seen. Just mind blowing & motivating.

Great video, especially all the specific examples. Keep up the good work!

Love your videos. They've been helpful and give a good overview of the STEM subjects, though may I suggest you do videos on Data Science. It's currently one of the biggest trends now.

I was looking for a video like this my whole life! Thank you!

Ah, this gives me hope! A long time ago, I set out to find the next even prime number. We know about 2. But are there more? Can we be so sure? So I applied for a grant, and another, and asked for donations from naive people. I couldn't get grants or donations from smart people. They just laughed and called me stupid! Stupid, my foot! I tell you, just because the first hundred prime numbers from 3 onward are all odd, doesn't mean you can jump to wild conclusions! I kept pleading my case for more research, for more budget, but they laughed. They LAUGHED! Well, they laughed when I came out with my new line of chocolate teapots. Okay, I admit that business failed. It was kinda silly. Never mind that. And then they laughed when I came out with a peanut butter based mouthwash! They said it couldn't be done! So I said, give it a try, gargle some of this antibacterial minty fresh peanut butter. But... all they did was laugh. So brainwashed into conformity, sad. And back when water parks were popular all over the country, I came out with a sandpaper park. Again, they laughed! THEY LAUGHED!! Well, we'll see who has the last laugh... if those misleading patterns as described in this video exist, then, ladies and gentlemen, we know nothing! We *MUST* continue the research to search for the NEXT EVEN PRIME NUMBER!! [dramatically pounds fist on table] [mental note: always carefully look for and remove any thumbtacks from a table before making dramatic gestures involving it.]

0:12 Uh, no, it’s 23. What’s the pattern? Sum up all of the previous *digits*, then add 1.

Great to watch...should be shown in every high school class. Thanks. Cheers

0:09 I remember studying it in my book (grade 9 book) under the chapter "Proofs in Mathematics", and it was used as an example to demonstrate why proofs are necessary. Though my teacher never focussed on it, because it was an additional chapter.

Man, thanks for telling me about Borwein-Borwein integrals. Made me happy to be a mathematician.

Nice examples. Thank you!

Good job Zach. Thanks for sharing!

I legit answered 31 at the start, i knew 32 would be too obvious from the title.

Love this video. Makes the elegant point that if you don’t know the algorithm, you can’t make assumptions.

That's a great video man I'd love to see more examples of this

The beginning of the video has a beauty to itself. Veritasium just made a video about 3x+1 where he said we are at 10^68, and in your video, 2 years earlier, we were at 10^18. Just nice to see the progress.

Statement: "Mathematicians need proof before they believe anything" Proof: This video References: 6:26

It is truely fascinating. It's things like these that I find really mystical about math. Indeed, why should such an integral fail after *that* many terms? It is far too easy to stumble upon patterns and simply will them away as fact. I admit, I do that myself too lol.

The shortened form of the "polynomial" is nC4 + nC2 + 1.

0:14 I got it right, but only because I expected some bs and guessed correctly, not because I found the pattern.

Great video. Thank you.

Hi great video! I have my own interesting case of a hidden gotcha from university. Me (maths degree) and my friend (Computer Science Degree) had a shared course in the first year. We were given a problem to explore using programming. It was to find the simplest way to calculate the power of a number. On first inspection it seems like for a number x^b you just multiply x by itself b times. But if you consider an index of the for b=2^k the answer can be found by continually squaring your last answer. In fact we only need to consider the indices of the base as x^s * x^t = x^(s+t). The problem can be rewritten as a sequence of summations of indices: a_0=1 a_1=a_0 + a_0 ... a_i = a_j + a_k, where j

Very very interesting dude! Great content :)

That was really excellent, as always!

6:03 How the hell can ANYONE assume this will produce prime number? As I remember didn't Euler assume that? Even a newbie can see if we apply 41 we have 41*41 + 41 + 41 = 41 * k wtf

2:28 such a mood hahahaha

Your trig integrals that supposedly equal pi over two are missing bounds.

I'm not a math major, so can I ask why do these "patterns" break down at such high values? It seems strange.

Cool Video! What do you have on your desk? A levitating globe?

That was a nice video, thank you!

You really put that number in perspective

The collatz conjecture is best solved in base 4. similarly in base 10, multiplying by 9 and everything will 2-cycle, as 1->0->1, 3->8->3, 5->6->5, 7->4->7, 9->2->9, if the rule is still to divide by 2 then we get that 1->5, 3->1, 5->3, 7->1, 9->1, but all of which over-all grew. in base 4, we get the final digits 0,1,2,3: 0 can divide by 4, 1 *3 +1 = 4n, and divides by 4. 2 divides by 2. 3 *3 +1 = 2n, divides by two, and needs more work. here we see all digits but 3 necessarily shrink, and we only need prove that all 4n+3 eventually get smaller, even slightly, since it will either get to 4n, 4n+1, 4n+2, or a smaller 4n+3, and therefore converge.

Real talk I completely forgot your name used to be MajorPrep. I love the name Zach Star so much more!

Hey zach i love the way you teach on you tube. 😍😍😍😍😍😍

Would the equation for the problem at 0:50 seconds look something like N x 300% +1/n = (50%) y Given n is the starting number and y is the number of evens you could divide. Then you could solve algebraically to get some not whole number that could potentially be in a different repeating loop with itself?

Using CorelDRAW, I played with several symmetrically placed dots on the circle, and get a result of 30 for n=6. Moving one of the dots away from symmetry creates the 31st region in the middle.

I just changed my major to math :) so hyped up

This is such a good video!

Love your content bro

I feel I bit too proud for knowing the 31

Why don't a lot of people watch ur good content???!!!! this stuff is so awesome...

In general, similar integrals have value π / 2 whenever the numbers 3, 5, 7… are replaced by positive real numbers such that the sum of their reciprocals is less than 1. That is the general formula

What if the solution at the beginning is 7? That would be the pattern of "Double if single digit, otherwise add the sum of the digits". So it would be 1, 2, 4, 8, 16, 7, 14, 5, 10, 1 (repeating)

Great examples

@ 3:37 Its only 32 regions because they're not drawing the dots symmetrically. If you look at the triangle region marked by the 3 lines formed from each pair of OPPOSITE dots (the smallest triangle in the center of the circle) , you'll notice that that "triangle" would dissapear if the dots were equally distributed. = 32 regions

Now you may call it bias just because I love maths, but this channel honestly deserves way more subs!

Hey, I was just wondering if for one of these proofs with the misleading patterns you could try to apply mathematical induction as a proof. I always felt a little suspicious of this proof-method for some reason. Would you be able to detect an error even if the first off-pattern-number would be like 10^43 digits away? Would be interested on a video on this topic!

When you have a product of sines in an integral, you can always use complex exponentials to get it in terms of sines and/or cosines, just with varying frequencies. sin(x) = (exp(ix) - exp(-ix))/(2i), cos(x) = (exp(ix) + exp(-ix))/2. This follows from exp(ix) = cos(x) + i*sin(x). Once you have a sum of sines and cosines instead of a product of sines, the integral is trivial. This would take a decent amount of work even for the integral with 15 terms, but it'd be nice to see how it works out.

4:40 If you replace (17, 9) by (23, 6) then the string of 1's gets WAAAAAY longer. How do we know? Well, one can construct polynomial equations that list the counterexamples, and both the degree and the coefficients of those polynomials tend to get very large.

Good video with good examples. Proofs are almost life and death when it comes to encryption.

FYI, the Collatz Conjecture is such a mystery they had to use BOINC to do research on it.

Just saw the movie you talked about here, never knew about Ramanujan sad but inspiring story. He was clearly a genius who thought outside the box.

I just had to double check your math on that stack of 10^43 papers. Yup, got about 7,7 trillions. Cool ^^

Love it how we all need to study proof, logic, statistical rules Its like saying "before you can work in the field, you need to know the laws of the land" and it's really necessary, even in natural sciences

Polynomial is outing prime for lower n values because probability of finding prime is more in lower number. However when values gets higher its difficult to Find prime which can be related to log function

thanks for the video

The correct answer to the "1,2,4,8,16 what comes next question" is: "it depends". And this would be true for any finite sequence of numbers. I thought correctly that he would ask 31 because of a previous YT video I saw.

somewhere in the zero area are reminders... because before you count 1 apple, before it was 1 it was a plant growing from a seed and giving out a flower to get to 1 apple all the process before it was 1 apple it was zero...

1. It becomes algorithm when we divide number by 2 if it's even; and multiply by 3 then add 1 if it's odd. Any number even multiple of 4, who is not multiple of 3, will yield 1. For odd number, we are pushing it to even number; and if result is even multiple of 3 or odd multiple of 4, it will eventually yield odd. So eventually we are taking out even multiples of 3 and pushing number to even multiple of 4, which finally gets us 1. There are some even number which are common, multiple of 4 and also even multiple of 3, these yield odd. 2. For circle dots, number of regions is reduced because after 4 dots, while connecting dots, some regions may not get intersected. Placement of dots may affect number of regions. I think, if even multiple of 4, if that multiple number is also multiple of 3, then we get odd eventually, or else 1.

2:21 I am confused. The extra term here should approach 1. So you're multiplying by 1. How does the value change the more terms of 1 you add?

Mind-blowing. I am applied math master student. I thought that in any practical scenario you never need to check statements beyond N approximately 100.

That first integral is a special case of the integral of sinc(t)*sinc(a1*t)*...sinc(an*t). As long as a1+a2+...an

That would be the induction method and that is why it is said that it is not a true logical method. Thank you Major prep!

Are these findings sufficient to warrant discarding inductive reasoning for numbers? Cus it seems that in the realm of insanely big numbers usual symmetries cease to exist. What is your opinion on this?

To go one step further, the proofs also need to be understandable and make sense so it can be verified. This is why many computer-generated proofs aren't considered proofs and why IUTT still isn't really accepted.

As soon as I saw the title, "Misleading Patterns," and saw, "1, 2, 4, 8, 16, ?" I immediately thought of the _3blue1brown_ episode, "A Curious Pattern Indeed," and thought you must be going for 31. ru-vid.com/video/%D0%B2%D0%B8%D0%B4%D0%B5%D0%BE-84hEmGHw3J8.html

Wow! I did not know about that sine integral, same result until getting up to around 10^43 - that is weird! I mean, WEIRD!! Great animation and explanation. This should convince anyone not to jump to conclusions about Goldbach, Collatz, or the Riemann Hypothesis, or any other such thing, no matter how many cases have been tried numerically on computers, even if tried in an airtight way to ridiculously huge values.

My next question would be at which point do these equations go from breaking and not following, to going back to working. Just cause it doesn't follow it once doesn't mean it wont follow again?

There's also the problem in Euclid's (proposition 1???), that if you draw two intersecting circles, they don't actually intersect.

With the Colatz conjecture, the interesting numbers are the odd numbers. This is because an even number will always reduce to an odd number that is smaller than itself -- n/2 < n. The next number for an odd number n is 3n+1 from the rules of the Colatz conjecture, which is guaranteed to be an even number. That even number will be divided k times for some value of k such that it reduces to an odd number, so the even number is reducible by 2^k. Therefore, the next odd number in the chain is (3n+1)/(2^k). The pairs of odd numbers (n, (3n+1)/2^k) are (1, 1), (3, 5), (5, 1), (7, 11), (9, 7), etc. Therefore, a different way of stating the Colatz conjecture is that (1, 1) is the only loop in the graph of odd numbers. Another interesting thing is to look at the values of k for each odd number (i.e. how many times you need to divide 3n+1 to get the next odd number). That sequence is: 2, 1, 4, 1, 2, 1, 3, 1, 2, 1, 6, .... I have tested this upto n=599 and every other value of k is 1. That is, k=1 for the numbers 4x-1 (3, 7, 11, 15, ...). I'm not sure why this is the case, nor if there are any other patterns. It would be interesting to see what patterns there are in these k values and how they relate to (n, (3n+1)/2^k). A result of this is that the numbers where k=1 will increase (as (3n+1)/2 > n), while those with k>1 will decrease (as (3n+1)/4

There's a running joke in my Geo class, if there's something we don't understand, we say it's because of Segment Addition Postulate. It's way more funnier in person.

In the movie The Man Who Knew Infinity there is an exchange between Ramanujan and Hardy that few would understand. R. thought li(x)2 but Littlewood had proven there are infinitely many positive integers n for which li(n)>pi(n) and also infinitely many with li(n)pi(n) for some n

I didn't think proofs were that important a few semesters ago. I used to say "you don't have to prove it to me, I believe you".

For the first problem are you able to use non whole numbers?

Thanks

People often forget the basic thing about "Continuity" before defining the Function.

I’m pretty sure the reason 6 points make 31 segments instead of 32 has to do with how obtuse the angles of polygons with 6+ sides become. A triangle starts with an average of the acute 60, squares are perfect 90, pentagon 108. Then when we get to double the triangle at 120 with the hexagon it just breaks and we stop getting segments in powers of two.