Ben Sparks must be a great math teacher. Energy, big smile, almost childlike wonder.

The mod9 being equivalent to digital root is insane to me, despite being such a simple proof

I love that Brady immediately tried to make sure it wasn't just a property true of all primes! Very easy thing to miss, he's clearly used to mathematical thinking now.

What you could do is show 3 by example. Say "Pick two twin primes, for example, 3 and 5. Multiply them together to get 15, and we calculate the digital root, and we get 6. Now I'm going to make a prediction. Choose another set of two twin primes..." You've subtly eliminated the bad case, AND you have shown that you can get different results. ;)

This is a nice way to check if two known primes are twin. No, wait-

"Proof by thinking" is now my favorite kind of proof.

I love every episode of Ben Sparks on Numberphile.

If you "want the 8 to appear" naturally, define your primes as 3k+2 and 3k+4 and multiply out to 9k^2 + 18k + 8. Of course, the demystifying bit is that generated pairs that AREN'T twin primes also yield 8 as a digital root anyway. For k=7, 23 x 25 = 575, 5+7+5 = 17, 1+7=8.

An informed numberphile viewer might remember another video, "Squaring Primes" from Matt Parker here. And indeed,in both cases, the first proof was an exhaustive one, while the second proof was the "prettier".

Not only is the number between twin primes divisible by 3, it also has to be even, so its always divisible by 6.

The algebraic proof also shows why 3, 5 doesn't fit the pattern. Since (n-1) = 3 in this pair, the conjecture that n = 3k is wrong. Neat!

The second proof tells you that the product of *any* pair of numbers that are either side of a multiple of three is congruent to 8 (mod 9). E.g. 14×16=224=8 (mod 9). You get the twin primes result as a bonus since they are always either side of a multiple of three.

I like Ben Sparks the best among all the brilliant mathematicians who appear on this channel. I still cannot forget the chaos theory video and the one with the mandelbrot set from Ben. Thank you.

You can use the same proof to prove that all cousin primes (4 apart) will have a digital root of 5, and sexy primes (6 apart) greater than 7 will have a digital root of 4

got a bit overexcited, I thought we had gotten a twin primes proof this is still weird and cool though!

Expect the pair 3 and 5 ofcourse!

Man, I've loved modular arithmetic ever since I first learned about it. It opens up so many unexpected doors in maths.

That was lovely. I love primes. More primes. The "3" between twin primes was awesome

always love the ben sparks episodes, i knew some of the ideas behind it but the way he puts the information is always so interesting

You can make this even more impressive. The product of two twin primes is also actually one less than a multiple of 36. You can proof this by doing the second proof and realizing that twin primes are always 1 mod 6 and 5 mod 6, which I believe was a numberphile video as well.

I like the little fact that the even number between twin primes is always divisible by three. Such a small thing but it means you can cut out two thirds of your search space when looking for twin primes.

This was so much fun. In uncertain times, it’s a relief to find that numbers remain reliable.

The second proof showed that although "Brady's Conjecture" was false, it has a cousin - that the digital root of the product of two integers either side of a 3 is 8 - was true. I think that although the second proof is neater, it does also highlight that it's not *really* to do with them being twin primes but rather them hugging multiples of 3.

This bring back so much memory to me. Modulo arithmetic was fascinating to me when I was in middle school.

12:05 proof by thinking is the best type of proof

12:06 "We proved by thinking..." Is such a great line!

Very well presented. Thank you.

The 2nd proof shows that for any number n that is divisible by 3, the digital root of (n-1)*(n-+) is 8. None of of these numbers need to be prime. For example, n = 15 or n = 21.

Twin prime: the composite number between them is divisible by 6. add them together and they are divisible by 12. (cousin primes' composite is divisible by 3).

If 'p' is a prime number bigger than 3, then (p^2) -1 is always divisible by 24 with no remainder. I think numberphile has touched on this before, but this video has inspired me to investigate if another Modular arithmetic fact is available along that line if thinking 🤔

*Simplest proof* Twin primes greater than 3 are around a multiple of 2 (even number in the middle) and 3 (one of the (n-1), n and (n+1) has to be 0 mod 3). Therefore, the *general formula of a twin prime pair* is (6k-1) and (6k+1) starting up with k >= 1 (5 and 7 are first such twin primes > 3). Their product = (6k-1) x (6k+1) = 36k^2 - 1 = 9 x (2k)^2 - 1 = 9 x [ (2k)^2-1 ] + 8 = 9 m + 8 starting up with m = (2k)^2 - 1 = 2*2 - 1 = 3. [ starting at 7x5 = 35 = 9 x 3 + 8 ] Therefore, *Single digit root of twin prime = (6k-1)(6k+1) mod 9 = 8*

Nice way of teaching. Thanks for making numbers easy for us.

So this actually proves that the digital root of the product of two integers, prime or not, separated by two is always eight if neither of those two integers is divisible by three.

Great video and guest.

This was very inspiring. I'm a teacher as well, and especially version one of the proof is accessible for students who don't know a lot of algebra yet. I like it a lot.

Loving the Klein Bottle on the shelf behind him. I have one too 😁🤙🏻

In Portuguese, the digital root operation is informally called "noves fora" which means "nines out". 😀

So you can pick any integer divisible by 3 and take the 2 numbers on either side. 123,456,788 and 123,456,790

As my old (white-suited) mechanical mathematics teacher always repeated after setting up the mathematical model of a physical system; "and now we just let the algebra do the donkey work." That part of me prefers the second proof too. State the problem accurately, and just watch the solution fall out. It's magical; like it's just meant to be there. (Which, of course it is!) But now, as a software developer, I like the brute-force "try everything" approach a bit more too. That's more like; "This will get an answer. Your equations may have missed something, but we can be sure this covers all cases. Because we actually look at all of them!" For the proof in this video, of course, the distinction is mostly irrelevant. But I'm pretty sure that NASA would use the first approach to send people into space, whereas the Met Office use the second to give you a weekly forecast. (Even though that sounds totally the wrong way around!) Sort of. Point is; a very interesting topic, and the main thing communicated (hopefully) is that the proof is as much about communication as it is about fact.

Amazing and amazing how easy it's actually is. Started instantly some research by my own and thougth about other number systems with different bases than 10. The digital root for numbers of any integral base is modulo(base-1). For some bases you got simply a different value for the digital root and some bases like 8 have several digital roots for this problem. For base 8 for example it is 1, 3 and 6. Base 7 it's just 5. Base 7 is interesting in an other way, too. The twins are always 1 and 5 modulo to 6. The gap between them can only be 0 modulo to 6. So, the number between prime twins is not only always divisible by 3 it's divisible by 6 either, but this is pretty clear, because the number between the twins has to be even, too and so it's 6 aswell. (6k-1)(6k+1)=36k^2-1=mod9=8

Another banger video, Brad !!!

Twin proofs for twin primes? More like "Every time I watch Numberphile I always have a great time!" Thanks again so much for making all of these very high-quality videos on so many different topics within math.

You know...I think I need to learn how to write proofs and videos like this are going to help get me there. Thanks.

"... just kidding, I wrote a different number under each corner of the paper. Gotcha!"

Thinking about how digital roots work (I used to love digital roots at the age of about 8), in base x, the digital root of any number y would by y mod (x-1). This works in all cases, except for binary. So if you were to work out the digital root in hexadecimal of 173AB (just a random value), you would have to work out 173AB mod F, which when converted to decimal becomes 95147 mod 15, which gives us 2, so the digital root of 173AB in hexadecimal would be 2. Just another to do it so we get an answer that's not 1-9, lets do 10D. 10D mod F in base 16 is the same as the decimal expression 269 mod 15. 269 mod 15 = 14. 14 in decimal is E in hexadecimal, so the hexadecimal digital root of 10D is E (although that one could have been solved by simply adding 1 and D together).

So to reduce a number mod m, all you have to do is convert it to base m+1, find the digital root, and convert back!

That was excellent. Thank you.

The moral of the story, to me, is that this is a property of any two numbers on either side of a multiple of 3, and twin primes (other than 3 and 5) are a subset of those pairs of numbers.

(Here is an algebraic proof to the Collatz conjecture) The method to be applied has only two rules: - 1. For odd numbers multiply it by 3 and add 1 2. For even numbers, we keep on dividing till we get an odd number (if it's not 1 we apply rule 1 again) Now, for all powers of 2, we keep applying rule 2 we will eventually get to 1, which is 2^0. (So it has to be true for all powers of 2). All numbers which are not powers of 2, are either odd or some even number which is the product of some power of 2 and an odd number (other than 1). Now, if it's an even number we keep applying rule 2, to get to that odd number (other than 1). So, in both cases we will hit an odd number other than 1. Now, all odd numbers can be written in the form 2x + 1, where x is any whole number. So, when we apply rule 1 to it, we are actually converting it to an even number of the form 6x + 4. (3 multiplied to an odd must give an odd number, since 3 is odd. 1 added to an odd number is even. Since the number we started with was having the form 2x + 1, the even number formed by applying rule 1 will have the form 6x + 4.) But, when we apply rule 2 to this even number, the algebra tells we will never get the earlier odd number back again. This is because 3x + 2 > 2x + 1(unless x is 0, in which case 2x + 1, would have been 1). Even if, 3x + 2 is an even number when we divide it by two all the numbers that we get will be less than 2x + 1. Which means we will get different odd numbers every time. All these odd numbers, will have the form 2x + 1, but the x must have a different value for each one of them. This in turn means, the even numbers that we will get by applying rule 1 will be a different number each time. This process will continue until we hit an even number which is a power of 2, which will take us back to 1. Therefore, all numbers on which the method is applied following the rules will eventually take us back to 1. Hence proved 🙂🙂🙂

you can go one further, since the middle number must also be even, the product of twin primes is one less than a multiple of 36 (6k+1)(6k-1)=36k^2-1

I noticed that the digital root of the product primes with a difference of four seems to come out to 5 each time.

It is prime time for twin primes

Thanks to the second proof, we know that the product of twin primes will always be one less than the square of the number in between. that's kinda cool

Every time I watch one of these, I feel obligated to figure out the proof before we arrive at it. I ended up with the algebra proof and was satisfied, but humorously, hadn’t even considered just checking the cases mod 9!

Absolutely mind blown

I've a question about the mod 9 digital root proof: The digital root of 9 is 9, but 9 mod 9 = 0. Is it treated as a special case or am I grasping it wrong?

@0:28 I like his honesty

I'm 64. The algorithm suggested this for me after I had been researching my dyscalculia. 🤣 I really enjoyed the video though. This guy must be a great teacher.

Twin primes are of the form 6n-1, 6n+1. Multiply then gives 36n^2-1. Digit sum of 36 is 0. 0 times n^2 digit sum is 0, so digit sum of 36n^2 -1 is -1, which is 8

Such a great teacher

Brady, long time viewer first time commenter here, love the channel👍( also from Adelaide) in a unrelated topic i think it would be really awesome if you could do a video about how to visulise very large numbers. i think it is(for me anyway) hard to understand how vast some of the numbers are, even a number 100 digits long is so massive. it would be awesome to have some visulisation on this. Thanks for the awesome content👍

I miss my twin brother, he passed several years ago 😢. I'll see him soon.

Excellent video.

Great video. Ben is slightly mistaken at 13:05. As he showed, the product is 9k^2-1. To get +8 you have to rewrite it as 9(k^2- 1)+8

Thanks for giving mr a solution to my question

Very beautiful proof

Is there anything interesting about the amount of steps involved in the summing to arrive at the digital root?

The number between twin primes is a multiple of 6. Thus, the mod9 of twin primes product comes to 8, because the product is ((6x)-1)×((6x)+1)=(6x)^2 - 1; (6x)^2 is a multiple of 9; subtract 1 and get 8.

The second proof makes it clear that this is a property of any two numbers which are one more and one less than a multiple of 3, prime or not. For example 20 and 22 have this property.

He's the best math teacher I've ever seen

Would love to see Brady’s reaction when he’s involved in the questions. Would be awesome

The product of twin primes > 3, is congruent to 8 mod 9. Is this true of any twin primes (other than 3 & 5, of course)? Yes, because it's true of any pair of integers that bracket a multiple of 6; and all twin prime pairs do that. (6k+1)(6k-1) = 36k² - 1 = 9(4k²) - 1, which is congruent to -1, and therefore, to 8, mod 9. The reason twin primes always flank a multiple of 6, is that any pair of integers that differ by 2, must include a multiple of either 2 or 3, unless they are ±1 mod 6. Which is also why (3, 5) are exceptions to the twin-prime-product rule. (A multiple of 3 can't be prime unless it is 3 itself.) Fred

Interesting in that the number 8 has always, like ALWAYS, been my favorite number.

I see our resident "Maximus Mathmaticus" is back...and yes we are entertained!

Awesome stuff gentlemen. :-)

The product of twin primes (excluding 3 and 5) is also equivalent to 35 (mod 36). Proof: All twin primes other than 3 and 5 are in the form (6n - 1, 6n + 1) where n is a positive integer. (6n - 1)(6n + 1) = 36n² - 1, which is equivalent to 35 (mod 36).

Thanks!

I think I want to see if I can work out similar proofs for other bases... My guess is that not every base has an equivalent proof, but I can't say why yet. Should be fun!

I play with primes alot myself and have found something amazingly simple and answers many questions about the "randomness" of primes. If you'd like to see it, message me.

Wonderful!

soooo so awesome!!!

The product is not only always (except 3*5) 8 mod9, but also 35 mod36

Yes, if 2 is the difference between two primes (unless one of them is 2 or 3), ie if they're twin primes, then the product of their digiroots is always 8. (Bearing in mind this applies to other number pairs of the form 3k+2 and 3k+4). If the difference is 4, such as in the example given in the vid, 7 and 11, then the pd is always 5. If 6, then there 's no single pd since 6 is a multiple of 3. The pd's are 1, 4, or 7. If 8 then 2, if 10 then 2. 12 is another multiple of 3, so 1 4 7 again. 14, it's 5, 16 it's 8 again. So a cycle of 8 5 2 2 5 8. If the difference is 0, such as between 13 and 13, then since 0 is a multiple of 3, then it's 1 4 7, which are the digiroots of all perfect squares except for 3 and multiples.

I did 59 and 61, the product of which has a digit sum of 26. Then I found 4 digit twin primes and their 4 million + product... also had a digit sum of 26. ! PS I've just in the last couple of weeks been experimenting with the Goldbach Twin primes conjecture (every even number larger than 4208 is the sum of two twin primes - that's primes that happen to be twins, not twin prime pairs themselves), and I found out a lot of things about twin primes, but I never suspected that every digit root of their product would be 8. I'm now going to watch the remaining 13 minutes of this video and find out that I'm being an idiot. Edited: 9:15 Yay! This was the first proof I ever did on my own (and in base k digital root of any number n is n mod k-1)

my favorite twin primes? Already covered that in my mega fav number video :) 8675309 and 8675311 are twin primes

The 2nd proof also says find any multiple of 3, the product of the number above and below have digital root 8, prime or not.

Long time no see, Ben!

You have done some really cool videos. What you don't answer with many - if not most - is the why? As in why is this cool or important. What greater issue does this lead to, or help solve?

Wow, I would really like to say "thank you" for this extraordinary prime number video. Why? Because I like to see proofs for very non-obvious things, and here we get TWO proofs for a very non-obvious thing, which is actually really exciting.

7:05 is it just a coincidence that these perfectly match up with the decimal digits of 1/7 (1, 4, 2, 8, 5, 7), but in a different order? or is there a reason for that behavior?

Twin primes (except for 3,5) must be separated by a multiple of 6 ; more generally, by a multiple of 3. So (3n - 1)(3n +1) = (3n)^2 - 1 = 9n^2 - 1~ 8 (mod 9).

I love Ben Sparks

While I don't question the results I do like to think about the potential for stagecraft trickiness - the "well I already wrote my prediction so it doesn't matter if I see see the calculations" thing *could* potentially be used to trick a mark in to making any prediction work, at least for a single game, since once you're watching the calculation could just give arbitrary operations on the fly until an opportunity arose to give one that reaches whatever "prediction" you wrote down...

There's another infinite sequence of twining: Step 1: Take a prime pair. Step 2: Multiply those. Step 3: Subtract that until you reach a twin prime. for (3, 5), I did: _(3, 5)_ (11, _13_ ) (137, _139_ ) and ( _18_ , _919_ , 19,079)

Digital root being 8 is not the exclusive property of twin primes, any numbers of the form 6n-1,6n+1 would work(specifically twin primes are in this form)

12:15 Awesome proof!

Does the second proof also show that the phenomenon is not limited to twin primes, but for (at least) any n where n=3k ? (For k, a positive integer)?

Twin primes? Exciting

It would've been nice if Ben had a twin and they did the twin proofs for -seven brothers- twin primes

This is the same kinda mental wavelength to me as when Matt showed the prime/24 connection