I love these classic Numberphile videos where James Grime talks about one particular number

Probably my favorite guy on Numberphile.

I can feel 7825. When I go somewhere where everyone is having a good time, everything stops when I get there.

Interesting how 7, 8, 2, and 5 were the digits for 7,825 which also gave you the original issue in a+b=c.

That blue pen appears rather green

I always get excited when James grime is on numberphile

*_lol blue and red makes purple, so 7825 is purple, silly mathematicians_*

Ramsay theory : The formula to find lamb sauce.

I have to do some calculus exercises... Me: "Okay, right now I'm absolutely not in the 'math-mood', let's just do other things first." *Numberphile just uploaded a video* Also me: Let's just watch this...

Absolutely loves James Grime, always a pleasure to see him featured on Numberphile again.

Fine example of a Parker Grid at 2:31 there James

James Grime appears in my subscription box: instant view.

...and those two days of supercomputation probably cost more than 100$ :)

"imagine all supercomputers in the world checking all the posibilities since the dawn of time you still wont be able to check all the posibilities ... so they used some clever mathematics to reduce the number they had to check, and it took them about 2 days" i snorted when i heard this, but this is actually great!

I bet I could do this for a^3 + b^3 = c^3

Whenever there are videos with James in it, I never hesitate to see it at that instant.

For the a + b = c case, there are really only 3 options, not 512 - either 1 and 2 are (say) red and 3 blue, or 1 and 2 are different colours and 3 is red or blue. The other colours are derived from there until it becomes impossible.

I have synaesthesia only where numbers are concerned. As a child, I was trying to find a way to make the numbers add up in such a way that their respective colours would blend to give me the correct colour for their total. I quickly ran into the same problem you guys did.

"The number 7825 must be both red and blue, which it can't" Superposition: "Bonjour"

it doesn't look blue because it's on brown paper.

Always enthusiastic, always interesting. So glad James Grime does these.

this problem reminds me of the 17 sudoku clues problem

Cool! I just learned how to come up with pythagorean triples. One of the side affects of watching numberphile: you might accidently learn something.

This problem reminded me of a project I did five or six years ago in university, which was also about finding sets of numbers without a certain structure - in our case, we studied Szemerédi's Theorem - and we were using SAT solvers to do this. Turns out that this work is from our supervisor at that time - Marijn Heule was our supervisor. And, indeed he used SAT solvers to create this proof. A little background: SAT this is the satisfiability problem, which asks "given a set of boolean variables (which can be either on or off) and a set of constraints on these variables, is there an assignment of values to these variables that satisfies these constraints?". You can translate the pythagorean triplets problem into SAT in its most basic form by creating a variable for each number from 1 to n, and creating a constraint for each pythagorean triplet stating that of the three variables corresponding to the three numbers at least one must be true (red) and at least one must be false (blue). If there is a solution for this SAT problem, then there is also a solution for the original problem, by colouring the numbers red if the variable is true, and blue if the variable is false. The main advantage of this approach is that SAT is quite a well-known problem, having quite efficient solvers. I can imagine that someone must have created an efficient SAT solver for supercomputers, or at least the Texas supercomputer, as well. This way it would be relatively simple to tap into the huge amount of computing power available in an efficient way (which is really hard if you're doing that from scratch). If I'm making it sound easy: it's not. The basics are quite easy to grasp (and can be coded in 30 lines of C), but in order to get an answer for these huge problems in a useful amount of time, you need to optimize your translation. This requires thorough knowledge of both the solver and the problem, to translate the problem in a way that is most meaningful to the solver. And finally, you need to handle all the data and include it in your proof.

Aaaaahh! I've missed seeing James!! 🤗🤗

*625² + 7800² = 7825²* *5180² + 5865² = 7825²* WTH, how will we construct a right triangle with this?

Number: **exists** Numberphile: Do you have a problem?

I laughed so hard when he so calmly said "so they reduced the number of scenarios they had to check to about 3 Trillion" XDD

5:30 Actually Ramsays theory allows you to solve for the location of the lamb sauce .

how he is amazed about what he's talking about, it's contagious :)

One of the most fascinating videos you've ever uploaded ! Great one !

This is probably the least click-baity title I've ever seen. That's something I really like about this channel.

Then proving that, for n colors, there will always be a perfect square which can be written as the sum of n other perfect squares and that those pairs must be the same color will prove the conjecture?

Great video, love that we're discussing a specific number

Ramsey theory has always fascinated me. Thank you so much for this video!

Where have you been James?! My favorite numberphile guy! Glad to see you back! J

I almost never understand this channel, but I love it anyway

The time estimate for a brute force approach is a bit off, because you don't have to check all of the possibilities. As soon as you come across a contradiction, you can stop and go on to the next set. Similarly, you only actually have to check 4 of the “512” combinations for the introductory problem to prove it never works (and indeed if you expanded the introductory problem to an arbitrary number of integers, you would never have to check more than those 4).

Love watching your channel! It’s the never ending pursuit of beauty!

These videos have no expiration date. Enthusiasm for mathematics is everlasting.

I found a pattern where a^2+b^2=(b+p)^2 made pythagorean triples if p was a factor of a. (lowest factor being 2 if a is even) I noticed in the video that for 7825; the two triples that make it up one of the two follow this pattern, and the other does not.. Namely 5180^2+5865^2=(5865+1960)^2 .. obviously 1960 is not a factor of 5180 in this case whereas 625^2+7800^2=(7800+25)^2 is. It could be nothing, but.. This was the first time I've come across a case where the a pythagorean triple exists that doesn't fit the above format. (p not being a factor of a). Perhaps this is why?

If there is always a number you'll reach where these equations break once you count high enough, wouldn't this calculation reflect entropy? With entropy, systems lose organisation over time. With these patterns, the number you reach before being forced into a 3-of-a-kind pattern gets higher as you add more colours. So wouldn't it be possible that this equation could end up having relevance to the concept of entropy if and when we solve it?

That blue pen didn't look very blue to me

I love that James is still a guest on this channel. I think he was in the very first Numberphile video. That was like 5 years ago, wasn't it? Holy mackerel.

Furthermore, I notice that Mr Heule has been working on Polymath 16 (the Hadwiger-Nelson problem) and just broken his own record for the smallest unit-distance graph with chromatic number 5.This follows the recent breakthrough by Aubrey de Grey, although I imagine this record may not hold for very long, considering the great progress now being made in this area.It's not so difficult to explain .. so maybe a future numberphile video please?

Shout out to the nail and gear in the background!

If you extend the case of sums of first powers to 3 colours (so we're excluding monochromatic triples (a,b,a+b), where a and b are distinct), then you can go at least as far as 22, using colour classes {1,2,4,8,11,16,22}, {3,5,6,10,12,19,20,21} and {7,9,13,14,15,17,18}.

4:32 omg I tried to find this for such a long time thank you!

I missed you Professor Grime 💕

What, a new Internet Comment Etiquette AND a new Numberphile video at the same time? is it Christmas already??''

He is just fantastic. Loves numbers. Loves math. Is completely consumed by his wonderful passion. And does his very best to spread the thrill and excitement.

If everyone had a teacher/professor as excited about Math as James then everyone would be a mathematician.

Can you talk about other patterns showing in the image? I see a lot of vertical lines.

7:52 - That's okay, you just have to defragment it and that will make the red go away. Or am I misunderstanding the problem?

2:50 He turns into Michael Rosen for a split second when saying "twelve".

Related trivia: Ramsey Theory you mentioned and a vaguely similar problem about graph colouring is where we got the gigantic Graham's Number from.

7825 should be either: 1. In a superposition of red and blue 2. Purple or violet

"It can't be red and blue at the same time" Violet: bonjour

I don’t know what this man is saying but his excitement sells me every time.

These are great videos, enjoy them a lot, I always wonder how things would change if using a different base, instead of base 10. how would the maths change

I would've loved to have you as my math teacher.

whats the name of the generalized version of this problem

Everybody knows *42* is the base number system used by the computronium running our reality [actually, it is _Quadragesimal_ (40) plus an independent "C.R.C." on _Binary_ (2)]. -> Knowing that *7825* is part of the equation leading to the "seed" used on this _particular run_ on the multiverse will get us closer to formulate the ultimate question.

Always happy to see James on numberphile😄

Great video, though I'd mention that the Babylonians were making lists of Pythagorean triples long before the Greeks!

It does strike me as a potentially useful step in understanding why this is the case in that having a brute force example may help mathematicians identify patterns that hadn't been tangible before.

I love Dr. Grime's enthusiasm.

James Grimes on Numberphile! Always a great watch when he is the featured mathematician.

And how can we use these results?

Wait so is 7825 the first time there is a c in two Pythagorean tripled that have to be of opposite colors

This vid popped on my recommendations and I'm first time watching this channel. Mby not the biggest fan of math but I absolutely love how passionate and happy this guy is when explaining all this :D

It turns out 9 is the smallest number for which the "a+b=c" coloring described at the start of the video is impossible. Here's one that works for 1 through 8: Color 1, 2, 4, and 8 red; color 3, 5, 6, and 7 blue.

Who else is here just because it has James Grime in it?

"I didn't do it. I failed." So this is a Grime Square™?

I love this guy. Great enthusiasm and explanations combined :) But also... Is that a CGP Grey Symbol in the background? :D

I'd be interested to see a graph theoretic approach; this strikes me as a variation on the many graph coloring problems.

Gotta love how briskly and casually he goes through the method of creating a Pythagorean triple. That could be its own whole video!

I wonder if this has something to do with the built-in limitations in DNA.

I really love the new addition to the James Grime space of Numberphile with the Nail and Gear flag

My aspie ears can’t stand the sound of marker on paper, but I love these videos!

Am I colorblind or is that blue green??

8:52 3.6 x 10^2355, for the record.

This was a fun video. I saw immediately that 1 thru 9 was impossible to not duplicate the same color. It happened for me when he showed the number 9 would be red. Bingo! That did it. Then, when he was discussing the 7825 number, I knew it was too large to ever examine every possibility. That somebody narrowed just a subset to 1 trillion was amazing, and then took 2 days to run the sets, was also thrilling to me. This sort of problem is what makes math so wonderful and inviting to me.

They tried to fit the second part of the truly marvelous proof in the margin, but it was too narrow to contain it.

Hey Brady, please resume making videos on the wordsoftheworld channel! Bring it back to its past glory

Even tho I'm a tenth grader, and sometimes I don't decipher completely what numberphile's guests are explaining, but I still love this chanel😍.

I appreciate the Nail and Gear logo in the background.

Not only is 7825 the sum of two distinct combinations of two perfect squares, but oddly enough, it's the difference of two perfect fourth powers: 13^4 - 12^4 = 28561 - 20736 = 7825

Me gustaria que hayan subtitulos en español, la verdad me gusta mucho este canal pero no se mucho inglés. Seria una gran ayuda para mi y para una gran cantidad de hispanohablantes seguidores de este canal.

These things are interesting, but what I would like to see in these videos is more explanation of why it matters.I know he talks about not knowing why it is true, but I want to know why it matters at all. Why does it matter if you can or can't color in numbers a certain color in a formula?

well this problem is also connected to fermat's last theorem as anything in the form a^n+b^n=c^n is part of the theory. If you noticed thought he addition of 1 to 9 also add the same issue as 7825. There are two sets of possible additions for 5 and 8 that break down the rules. 9 is another example of two pairs that are in opposite colors, so not matter what is chosen the coloring breaks. we might want to stick to the addition to understand this problem. multiplying is only causing us greater amount of work.

Is the same limit encountered in octal or hexadecimal or does it only apply to decimal calculations?

CGP gray print could not be unnoticed

The symbol in the picture in the background looks like CGP grey’s logo

Hey Brady, You shown us a lot of discrete value problems that have not been solved. Can you make a video about a discrete number problem that has a proof that comes in less then several terabytes?

is this the problem that aubrey de grey solved ? (not fully solved but removed the 4 probability)

We may not understand why it's true, but knowing that 7825 has this property is surely a stepping stone towards finding an analytical proof?

Nail&Gear in the background

It would be interesting to know how far you could go in reverse before it failed, i.e. start at 7825 and work backwards.

The best Numberphile is back