357686312646216567629137 - Numberphile 

Numberphile Numberphile wearing at t shirt with a prime number on it is now a thought crime in the US. Although your chances of being detected are slim (how many members of the thought police even know what a prime is, and how would they explain their knowledge without implicating themselves?). The penalty is to be sent to a de-education camp (trailer park flooded with meth).

How much is pinning this in the comments worth? You can keep the number in GBP.

Joshua Jansen 50 pence?

Numberphile About 6 mins in; OH WOW. Yeah.

Do you have to re-edit your videos if the company that's sponsoring them eventually stops the offer being offered?

try numberphile prime numbers

Thank goodness for the keywords/tags and the description, then.

Considering the importance of the number, all cultures will eventually find it.

what about the tom scott video written in inuktitut

Let's start searching for videos by just entering in truncatable primes in other bases, and see what comes up. Even if you don't use base 10 by default, you can still find the video if you expand your search to other bases, assuming you at least use a similar number system.

Same

Tantusar Just to further thid argument a bit: 2 and 5 must be the first digit because any number 2 digits or greater ending in 2 or 5 is non-prime. No digit can be repeated or you can end up divisible by 11. Since all the 2 digit numbers have a 3, you can't have another 3. You can't put a 7 on anything with a 7 and you can't put a 7 on anything starting with 2 or 5 or you'll get 27 or 57. You can't put a 2 or a 5 on anything with a 7 and you can't put a 2 or a 5 on anything with a 2 or a 5. That eliminates all the 3 digit numbers.

Yeah, I did some coding on Mathematica, those are the only ones in which you can keep deleting digits forever and only get primes. but I did find several which are primes for at least one deletion of a random digit, here are the results among the first 1,000,000 primes: 23, 37, 53, 73, 113, 131, 137, 173, 179, 197, 311, 317, 431, 617, 719, 1013, 1031, 1097, 1499, 1997, 2239, 2293, 3137, 4019, 4919, 6173, 7019, 7433, 9677, 10193, 10613, 11093, 19973, 23833, 26833, 30011, 37019, 40013, 47933, 73331, 74177, 90011, 91733, 93491, 94397, 111731, 166931, 333911, 355933, 477797, 477977, 633317, 633377, 665293, 700199, 719333, 746099, 779699, 901499, 901997, 944777, 962233, 991733, 1367777, 1440731, 1799999, 2668999, 3304331, 3716633, 4437011, 5600239, 6666437, 6913337, 7333331, 7364471, 7391117, 13334117 Remove any digit form these and they remain prime.

Ok, Now do it for any base.

First I thought, "of course they EXIST -- 37 is one. The question is, do LARGE ones exist?" ("They" being "arbitrarily deletable" primes that remain "arbitrarily deletable" all the way down the chain to one digit.) Then I immediately realized they could have no repeated digits, since if they did, you could delete all the other digits and get down to something divisible by 11. They also can't have any digits but 2, 3, 5, and 7 (the one-digit primes), so you can't possibly get beyond 4 digits. And if they have a 2 or a 5, it has to be the first digit, since any number that ends with 2 or 5, other than 2 or 5 itself, is not prime. That leaves very few possibilities to check, and checking all the 3-digit possibilities quickly shows that you can't even get to 3 digits.

Just worked that out before going through the replies. Kind of a disappointment that there's nothing beyond 2 digits that satisfies the conditions.

Perfect misdirection!

That's an interesting statement, but what if we said this about Matt? It would be true then, right?

(which is almost every video)

Numbers? You mean Noobahs!

Agreed, I used the same logic

wow, nice

That's true, I ran a program testing the first 1000 numbers.

i got the same answer for base 10. the list ends at 73 because no 3-digit numbers can give 3 2-digit numbers that are random digit remove primes which also chain reaction means no 4-digit numbers exist for this either. i want to see if i can go further in higher bases but cant find a list of primes in base 12 :/

In case you are interested, I made a little python program to calculate all random digit remove primes in different bases. I have not verified the code, but I am pretty confident it is correct. They are in their corresponding notation, using ABC for digits higher then 9. Interestingly, base 8 has a three digit number and base 12 has a four digit number! 2: 3: 2, 4: 2, 3, 23, 5: 2, 3, 23, 32, 6: 2, 3, 5, 25, 35, 7: 2, 3, 5, 23, 25, 32, 52, 8: 2, 3, 5, 7, 23, 27, 35, 37, 53, 57, 73, 75, 357, 573, 753, 9: 2, 3, 5, 7, 25, 32, 52, 10: 2, 3, 5, 7, 23, 37, 53, 73, 11: 2, 3, 5, 7, 27, 72, 12: 2, 3, 5, 7, B, 25, 27, 35, 37, 3B, 57, 5B, 75, B5, B7, 357, 35B, 375, 3B5, 3B7, 5B7, 35B7, 13: 2, 3, 5, 7, B, 23, 25, 2B, 32, 52, 14: 2, 3, 5, 7, B, D, 23, 2D, 35, 3B, 53, 5D, 73, 75, 7B, B3, BD, DB, 15: 2, 3, 5, 7, B, D, 27, 2B, 2D, 32, 72, B2, D2, 16: 2, 3, 5, 7, B, D, 25, 2B, 35, 3B, 3D, 53, B3, B5, D3. 17: 2, 3, 5, 7, B, D, 23, 27, 2D, 32, D2. 18: 2, 3, 5, 7, B, D, H, 25, 27, 2B, 2H, 35, 37, 3D, 3H, 57, 5B, 5D, 5H, 75, 7B, 7D, BD, D5, D7, DH, H5, H7, HB, 357, 375, 3D7, 3DH, 3H5, 57D, 5BD, 5D7, 5DH, 5H7, 75D, DH5, H75. 19: 2, 3, 5, 7, B, D, H, 23, 25, 32, 52, B2. 20: 2, 3, 5, 7, B, D, H, J, 23, 27, 2D, 2J, 37, 3B, 3D, 3J, 53, 57, 5D, 7B, 7H, B3, B7, BD, BJ, D3, DB, DH, H7, HD, HJ, J3, JH, 2D3, 3B7, 3BD, 3BJ, 3DB, BD3, BJ3. 21: 2, 3, 5, 7, B, D, H, J, 25, 2B, 2H, 2J, 52, 72, B2, H2, J2. 22: 2, 3, 5, 7, B, D, H, J, 23, 2H, 35, 37, 3D, 3H, 53, 5H, 73, 7D, 7J, D7, H5, J3, JD, 23H, 35H, 37D, 3H5, 53H, 73D. 23: 2, 3, 5, 7, B, D, H, J, 27, 2D, 32, 72, J2. 24: 2, 3, 5, 7, B, D, H, J, N, 25, 2B, 2D, 2J, 2N, 37, 3B, 3H, 57, 5B, 5H, 5J, 75, 7B, 7D, 7N, B5, B7, BD, BH, BJ, D5, DJ, HB, HD, HN, J5, J7, JB, JN, N5, NB, NH, NJ, 25B, 25J, 2BD, 2DJ, 2J5, 2JB, 2N5, 2NJ, 37B, 3B7, 5BJ, 5HB, 5J7, 5JB, 7D5, B5H, BD5, DJ5, HBD, J57, J75, JB5, NJ5, 2NJ5. 25: 2, 3, 5, 7, B, D, H, J, N, 23, 2B, 2H, 2N, 52, B2, N2. Edit: Typo in the code, 15 is not a prime lol. Fixed now.

Yep, the primes go on forever! The easiest way to see this is to just take the product of all the primes you’ve found so far, and add one. This must always be a new prime!

More like the new number is either prime, or it divides into a prime thats larger than the largest prime in your list, either way youve shown that theres always a larger prime than what you originally thought was the largest prime

James Jumper - lol, Dad Jokes: Numberphile Edition

But it wouldn't make a prime product line.

*ba dum tss*

But if the pencil is unsharpened it's pointless

A RU-vidr uploads infinite videos on their channel. First video is a minute long, second one is two minutes long, third one is three minutes long. He puts all of those videos in a playlist. A viewer finished watching all of them 5 seconds before he made the playlist.

​@@tiletapper4everis this real? What's the channel name

@@thoughtfulsoul3402no, it’s a joke about the sun of all natural numbers and -1/12. Just look up -1/12 and you’ll get several videos explaining it.

@@thoughtfulsoul3402it’s a joke based on famous 1+2+3+4… = -1/12 result

Thats a contradiction

Maybe there exists *very* truncatable primes that verify this haha

Eliot any truncatable number that doesn't use 8 as a digit would work

Oh yeah you're right that's not fun :'(

You beat me to this comment! XD

Have you verified that? ;)

Pros and pros, it has no cons...

@@XenophonSoulis You do lose a lot of functionality if you're very strict about sticking to powers of 10. It's nice having highly composite divisions instead, so you can talk about quarters and thirds of an hour for instance.

@@fastpuppy2000 You don't need thirds of an hour if they aren't integer multiples of the second in the first place...

Let's just scrap hours and minutes and use kiloseconds instead. Sure, it'll mean a day on earth is 86.4 ks long, but there's nowhere else where a day comes out to a nice round number of seconds either.

@@Roxor128 there's no reason we'd have to keep using seconds though. Use a unit of time that's slightly shorter and have 100,000 of them in a day instead of 86,400.

This is how we spell in french. It feels so unnatural.

How does it feel unnatural? A billion is a million to the power of two, a trillion is a million to the power of three, and so on... This makes the long system feel much more natural than the short system (at least to me).

Cedros It has no point, if you really think of it. Why use one latin number every 2 10 powers?

YipHyGaming - Minecraft Agario Cytus and more! Because we started counting on the second group.

Counting "million, millionard" is unnatural. That's like counting "one, oneard, two, twoard, three, threeard". The value millionard represents isn't a subset of million or anything, so why would it look like there's a much stronger relation? Also, I know that qunitillion is million to the power of 5 in short system, that's easy. I have no idea what that would be in the long system, to the power of ten? How does that make sense?

huh?

An eraser to go with the pencil, with 739397 written on one side, and 'prime for writing wrongs in your life' on the other side.

or just for idiots like me that sharpen their pencils on both sides at once

@@onetwothreefour3957 you got three hands or something?

Exactly my thinking except for the last step: All four three-digit numbers are not prime, because they are all divisible by 3.

weilgooglesproduktintegrationsstrategievoneinemexmicrosoftsuitandtiemanagerfritzengeleitetwurdeunddahervonvornhereinzumscheiternverurteiltwar

Cedros I think he was just listing the only numbers that fit the criteria, and he may have accidentally glossed over the fact that they aren't prime which is even more proof that it can't exist.

Yes, it felt easier for me, because I saw the 27 and 57. Checking if they are prime only turned out to be easy, as they are all divisible by 3. Otherwise the prime check would have been more complicated.

Basically 73 becomes the largest

Numberphile Deleting 7 would give 33 = 3*11 which isn't prime :-(

The rest are 37, 53 and 73.

The only possibilities are 23, 37, 53 and 73:.

3C Kitani 1 aint prime, yo

tommihommi1 Yeah, i know xd

73

23?

If you allow 1 to be prime, there are 20 numbers which work: 1, 2, 3, 5, 7, 11, 13, 17, 23, 31, 37, 53, 71, 73, 113, 131, 137, 173, 311, 317

Wait... what number is 27 divided by?

I love refrigerators

Love your name

I like trains

Why doesn't anyone like Easy Bake Ovens?

jake me Too

kiffe22 I’m confused too, 453 and 45 are divisible by 3 as well

kiffe22 The thing with the 'Deletable Primes' is that you can deliberately _choose_ which one you delete. If the number that's deleted is random, we can quite quickly find out all of them. Let's call them Random Deletable Primes (RDP) or whatever. Say a number is RDP. That means it is a Left Truncatable Prime, because deleting the numbers from left to right is a random possibility. For the same reason it is also a Right Truncatable Prime. Therefore an RDP is in both the Left and Right sets (RDP is a subset of intersection(LTP, RTP), for those that like notation 😜). Both these sets are finite. Hence, the set of RDPs is also finite.

He must mean you can delete any digit, but only delete once.

But that's not the way the example played out.

He's not saying that any number you remove will leave a prime. He's saying that you can try deleting any number, and if one of them is prime, you can continue to chain. Another way to say this is that at least one choice of digit can be removed and create another prime.

45, 56, 57, 63, 456, 453, 567, 573, and 4563, also aren't prime. The point with that example is you don't have to remove digits from the ends, but you do still have to remove them in a specific order.

That's pretty funny because mine is 357686312646216567629138(!)

After quick computations by hand, the list of such numbers is, 2, 3, 5, 7, 23, 37, 53, 73, 373 The largest such is 373

373 doesn't work if you remove the 7.

Yeah you're damn right, then 73

The restraint that makes this less interesting is that you can’t repeat any digit and you can’t have a 1 anywhere, nor an even number else you can end up with 1, or a factor of 2 or 11. That limits the whole thing to subsets and combinations of 3,5 and 7.

"hence there are finitely many and at least one, there's a biggest." , What makes you say there are finitely many of them ?

1 is *NOT* a prime.

Why pencil prime? That doesn't make sense. Truncatable makes sense because that's what you're doing, you're truncating it.

And if you're referring to the sharpening thing, that's definitely gonna cause more confusion for the people that doing get the reference. Besides that, that pencil was made after the logical name was already given.

Austin Bryan I just think it's a nicer name because it functions the same way as a pencil- you take some off the top and it still works.

Henry Bownes if you remove part of a pencil, you're truncating the pencil

Truncation is a standard name for the operation they're doing to the number.

Do you mean inauspicious?

There is no choice. We HAVE to have it. Put it in the shop, quickly, you're gonna make a LOT of money (maybe cause i'll buy a heck of a lot of them)

Roger Wang If you allow 1 to be prime, there are 20 numbers which work: 1, 2, 3, 5, 7, 11, 13, 17, 23, 31, 37, 53, 71, 73, 113, 131, 137, 173, 311, 317

He didnt say "random digit deleteable prime" he said deleteable prime. You can delete the digits in some order, not necessarily in any order.

Could including 0 lead to a potentially infinite number of these truncatable primes? For instance maybe 3000...trillions of 0s later...0007 is a prime so using the truncated method we would only need to check it and the number 7. Who knows how many of these types of numbers there are? You could then of course add any of the numbers found using the method in the video to the front of these other long numbers as well.

Zsemberi Dániel interesting, as the Video didn't include Zeros...

NJ S Yeah after writing my comment I gave it more thought and came to the same conclusion as you. Next time I'll think before I write :)

It would make for a perfect Parker's Truncatable Prime.

That did occur to me, but - at least from my point of view - it's a bit of a cheat. 7, 07 ,007 etc. are the same number, so you're effectively skipping a digit. A 30 digit number with 28 zeroes would only contain two different prime numbers.

2003

But then you can expect a number like 20000000......0000003 as a prime number, and then the number 357686312646216567629137 is no longer interesting.

One of the rules that they forgot to mention is that the number can't contain a zero for it to be considered left-truncatable, and you obviously can't have a zero in a right-truncatable prime because at some point you'd end up with a number divisible by whatever base you're using.

03 isn't a real number. You can't have 03 of something.

@@chrismanuel9768 03 = 3. It's just a matter of notation.

Nargis Akter then it doesn't say 'always in your prime' anymore either ;)

+ Nargis Akter: That's how you know when it's time to get a new prime pencil! Fred

at some point, it will say "ur prime".

Did you say +2 because Graham's number is an odd multiple of 3, so it can't be prime, and Graham's number +1 also can't be prime because it's even

They talked about this in the video. Larger bases create larger trees, meaning larger primes. However, it is unknown whether or not they will extend to infinity at some point, though I would guess they don't as prime density goes down with higher numbers.

How far does the sequence go in each base. I didnt pause but for base ten, it looked like less than say 200 left truncateable primes. I guess in base two the sequence is 11=3 111=7 and that's all. Just these two since1111=15 which is not prime.

That begs the question what is the maximal length of the sequence of left-truncatable primes as a function of the base.

Thinking about it a bit: - all digits must be prime themselves, so 9, 8, 6, 4 and 1 are out, leaving only 2, 3, 5 and 7 - no 2 digits can be the same or you can end up with something like 33 or 77 - numbers with more then one digit ending in 2 or 5 are not prime, one of them may be in the number but only at the very left side. None of 573 537 273 237 are prime.

... because they're all divisible by 3. Fred

bleesev2 Good proof. The only (non-trivial) fully deletable primes are 23, 37, 53 and 73.

Why not 317?

S-N-A-IL PS4 Remove the 3 and 7 to get 1 which is not prime.

You can also just look at the intersection of the left-deletable and right-deletable lists, since it will necessarily be a subset of that.

If you allow 1 to be prime, there are 20 numbers which work: 1, 2, 3, 5, 7, 11, 13, 17, 23, 31, 37, 53, 71, 73, 113, 131, 137, 173, 311, 317

You'd be adding digits, but you wouldn't always get a new number when you delte a digit. A 30-digit number with 28 zeroes would only contain two different prime numbers. And it would take a bit more work to test, since you could keep adding zeroes forever in the hope of hitting a prime number.

Yep that would be the idea and this list might not be finit because of that :)

Patrick BORE but it's already been proven it's finite, the fact that there's infinite numbers doesn't matter

Trimon The fact that there are infinite numbers does matter because there is an infinite amount of numbers with n amount of 0s and 2 digits that are prime. All of these numbers are probably prime. Also the fact that it is proven to have finitely many of these numbers have been proven to have finitely many without any 0s.

Jorge C. M. His IQ is prime.

And truncatable from any side as well as randomly.

so at most 73? That's pretty low....

At most? Are you crazy? How about 373?

373 is not truncatable randomly. If you truncate the 7, you get 33, which is not prime.

Joshua Hillerup it probably wasn't all done by hand but a program could fairly easily brute force it. Especially if you gave it a list of known prime numbers to compare each result to. Basically say that if the result is on the list continue, if not then try the next path.

You couldn't determine that by brute force because there are infinitely many primes. There must be a proof but I don't have any clue how you would do that.

adam poulter that's my assumption too, but I'm just wondering if that was done, or if there was a more elegant proof.

You wouldnt be able to prove a list is finite using brute force.

I'm pretty sure you can bruteforce that. Apply the same method he did with 7 => 47 => 947 => 3947. There aren't infinitely many paths.

T Perm, "In ANY order you want". Not one predefined order.

Hold on everyone, we need to specify between deletable and any-digit-deletable. It satisfies one but not the other. They were pretty clear about that in the video.

Chad Eichhorn, Ok, reconcile that with "In ANY order you want" for me.

Your _allowed_ to choose any digit (not being restricted to one of the ends) to create your chain of primes. There's no requirement that all choices must work. The existence of one sequence of choices is enough. I first had the same thought as you, but figured it out after a while. They could have phrased it a bit differently in the video. The existence of a 6 in the number is enough to throw the other interpretation out.

If you are allowed to remove a digit "in ANY order you want", what numbers are able to be prime after the removal of each digit down to just 1 digit. This is what he meant, not that all digits work all the time. Those numbers are 27, 37, and 57 and that's it none that are three digits or more.

Yes I agree....and especially 4 ist not a prime, so unless deletable prime is not quite the definition they gave us, this number does not fit....

A deletable prime must have all prime digits, as any digit could turn out last to be deleted.. This one has 4 and 6... a glitch in the (otherwise great) video.

another glitch: 4567 -> remove 7 -> 456 is not prime

He didn't explain it amazingly but a deletable prime only needs to work for one chain of deletions, not all posible chains. When he says "any" he means at each step we get to choose which one, not that every choice is correct no matter what. For the latter case the only numbers are 27, 37, 57.

S-N-A-IL PS4 1 isn't prime.

The numbers in hexadecimal: 2, 3, 5, 7, B, D, 25, 2B, 35, 3B, 3D, 53, B3, B5, D3, 2B3, 3B3, D3D. This doesn't work at all in binary because the only single digit numbers are 1 and 0. (If we counted 1 as a prime, the highest we could go is 111, the binary representation of 7.) I'm not sure what the significance of several of the three digit numbers being palindromes is. There may be a reason for this trend but I'm not sure yet.