Sup, Beardy. Can you go ask the russian math guy over there (cant remember who, tell me, someone alex?? younger skinny dude), why he mocked ppl that Collatz Conjecture is a "commie PsüÖp to hinder western progress". As it is (but not about the progress, it´s just satanism) and it can be proved by anyone who has learned basic math (basically any 6th grader).
any sum of twin primes comes in an [n, n+2, n+4] triplet so exception sets make sense but without watching the extra footage you mentioned i dont know why the exceptions would always be triplets, and not quadruplets or something
8:32 is absolute genius! I had not thought of that before Brady suggested it. I know we (rightly) praise the professional mathematicians on this channel, but Brady is a spark of light in his own right, and deserves much more credit than he gets!
I think you cannot proof the twin prime conjecture based on the twin prime Goldbach conjecture. Quite the contrary, I would think the twin prime Goldbach conjecture _assumes_ the twin prime conjecture is true. So I think the twin prime Goldbach conjecture would start with " _if_ the twin prime conjecture is true, …" or "Given infinite twin primes, …".
@@martinnyolt173 I don't see that as an assumption at all. It's a simple statement that is either true or false. Disproving the twin prime conjecture disproves the twin prime Goldbach conjecture.
This is what happens when you hang out with people who have more experience in a field (in this case, mathematics) than you do, but are willing to talk to and with you about what's going on rather than talk down to you.
Note: it is possible for both the twin prime conjecture and the Golbach conjecture to be true, but the Goldbach twin prime conjecture could be false. This is because the twin primes could get sparse too quick, that some massive number exists that isn't the sum of two of them. Think of all numbers that are the sum of two squares: There are infinitely many squares, and infinitely many sums, but there are also infinitely many integers that are *not* the sum of two squares. In fact, we already know in some sense twin primes do become sparse faster than primes: the sum of the inverses of twin primes is finite, but the sum of the inverses of primes diverges.
The fact that even after many centuries of trying, no one in the wold has discovered neither an intuitive or convoluted proof to such a fundamental and seemingly trivial question is why I love mathematics.
That would be such a human thing to happen. "Solve these problems seperately. " Humans: That's pretty tough... "Solve these problems at the same time!" Humans: Aight, bet
It definitely feels like 0 should be included as an even number that can't be written as the sum of two twin primes, so that it could form a triple with 2 and 4
I think it's the reverse: both 2 and 4 should be excluded. The reason for 2 is that the regular Goldbach states that every even number x >=4 can be written as the sum of two primes, so for Twin Goldbach we should also do x >= 4. And for 4, I would argue that 2 should also be on the list of Twin Primes. We could easily modify the definition and say that p1, p2 are Twin Primes if |p1 - p2| is *at most* 2, thus including 2, because |2 - 3| = 1
If you are just taking even numbers, they can be described as sums of unique powers of 2 (or more like, every integer can, but even numbers don't have 2⁰). Essentially taking the binary representation of the number. Summing two of them to get another number, is just summing more powers of two together, and removing that "uniqueness" as certain powers can be duplicates. You can easily reverse the process, take a binary number and do the steps backwards with literally any integer smaller than the target to get a second number you can sum that might or might not share powers of two. If you're summing two primes together, they're also just like summing unique (minus the duplicates they share) powers of two. So it's instantly a lot more realistic that you could get any number from that. As to why it's only even numbers, both primes are odd by definition and as such both have 2⁰, meaning their sum will NOT have 2⁰. The question then shifts: if you picked any even number, you could easily deconstruct it into two binary numbers. But if the conjecture is true, then there is ALWAYS a way to do so such that not just one but BOTH numbers are prime. Does this process always require shared powers? Because you can do so without shared powers (literally just split the target number in half high/low bits) but that doesn't guarantee the two halves will be primes. Could you even get two primes that don't share any powers (outside of 2⁰ which by default all primes must have)? Instinctively it feels like they need to share some, as some examples share all powers (namely ones where both primes are the same). It also links to Mersenne primes and related primes, defined as 2ⁿ-1, meaning by definition they contain all powers of 2 up to a certain point: can you make any even number by restricting yourself to at least one of the two primes being a Mersenne prime? That would by definition require shared powers (again outside of the trivial 2⁰), unless the second prime is either 2ⁿ+1 (the twin of the first) or at least 4 times the Mersenne one. Given the rarity of both Mersenne primes and primes in general, you'd need a lot of primes between the Mersenne ones, to be able to fill in for every even number, as such I don't think it's doable with this Mersenne limit. Such an interesting topic.
For a good example of why checking a conjecture up to huge numbers isn't a proof of that conjecture, read about Mertens Conjecture. It's been shown to be false with the first counterexample lying somewhere above 10^16 but below 10^(4*10^28). Perhaps unsurprisingly, no one's found an explicit counterexample yet.
I was thinking about primes yesterday, specifically prime bases, and wondering if James would be interested in doing a video about common properties of prime bases. I was thinking about prime bases on my walk home from work yesterday morning and I started wondering about whether there would be a set of properties that all such bases would have in common.
I'm not so intrigued by the fact that the exceptions come in triplets because I guess they're in "blind spots" that are just too far away from twin primes on both sides. But I'm fascinated by the fact that all of these triplets except one end in 4, 6, 8 and not 0, 2, 4 or 2, 4, 6 or 6, 8, 0 or 8, 0, 2. I'd love an explanation of that, and why there is one exception within the exceptions!
They should come back and shoot more footage. Maybe upload it to some secondary channel. And name it "extra footage", that would be fun. They could even put a link to it into description ;)
Here's something I was thinking about: Let's assume that Goldbacha conjecture ALMOST fails infinitely often, meaning that there exists large even numbers with only one way (up to order of summation) to write them down as sums of primes. Now, given that n is almost a counter example, how difficult (what is the complexity in log(n)) is it to find primes p,q such that p+q = n? Is it as exponentially difficult in log(n) as factoring semi-primes? Can one make an encryption method out of it?
Have we checked to see how many exceptions there are for the versions for cousin primes, sexy primes, and other gapped pairs? I expect the number of exceptions grows. But I wonder if there’s a point where there are infinite exceptions.
If one includes the differences between two twin prime numbers, many of these entries will fall from the list. Example: 4204 = 86927 - 82723. [86927, 86929] and [82721, 82723] are twin prime pairs. This is the only pair of differences for 4204. A quick calculation pares the list to [94, 514, 516, 518, 904, 906, 908, 1114, 1116, 1118, 1264, 1266, 1268]. Maybe worthy of inclusion in OEIS.
Cool. 1,1 = 2 (10 base counting system(binary)) 1,1,2,2 = 6 ( 32 or 12 counting system (trinary)) 1,1,2,2,3,3=12 (60 or 12 counting system( quad). Binomial to monomial ? You termed them up, making them prime. 1,1 = 2 for 10. 1,1,2,2 =6 for 100. 1,1,2,2,3,3, = 12 for 1000. It can not be 28 unless the counting system is not synced with the others or a greater counting system is present. Its effecting the smaller ones. Thats mostly what you are seeing. I'm on god level.🎉
The numbers that can't be made have the same relationship as sets of three even numbers that can be made by two pairs of twin primes. Every duo of pairs of tp add up like this: (sm=smaller pair bg=bigger pair) sm1+bg1 sm1+bg2 sm2+bg1 sm2+bg2 But sm1+bg2 must be equal to sm2+bg1, so there are only three consecutive even numbers that are made by the two pairs. The ones that fall between the cracks have to be related in the same way, QED!
One possibility is Goldbach’s Conjecture may be true but unprovable. It may just be one of those things which is true but we cannot know it to be true. Could this be proven?
The thing is, if the Goldbach conjecture (strong version shown in this video, not the twin prime or weak version though) is shown to being equivalent to an axiom of the system of math and numbers we use, wouldn't it then become necessary but impossible to "prove" since it would just be a result of the system we're using? I've always wondered if the Goldbach conjecture is impossible to prove (or disprove). The reason being this: What is the Goldbach conjecture? It isn't claiming that every even above a certain value can be described as the addition of two primes, it is actually claiming that we only need two primes. So, it *could* be described maybe with more than 2 primes, but the minimum amount needed is two primes. That's an important distinction because if we're trying to prove through contradiction we'd have to show the opposite.. so we'd have to show for at least one number value that it needs more than two primes added to be created. But, how would we prove that is or isn't the case unless we looked at every number and every amount of primes needed (so for example maybe some number way out there requires 6 or 8 primes to be formed. While I don't believe this to be the case, how do we exhaustively prove or disprove it? I don't think it is possible since we have to check infinite numbers at infinite varying amounts of primes needed to be added to form them. Does this *prove* that the Goldbach Conjecture cannot be proven? No. But, it does suggest it cannot be proven (even if it is indeed true). And, since I'd like to be the one to prove it (or at least be there when it is proven).. this possibility is a little depressing actually. Great video though, I almost always love numberphile videos, especially their topics and subject matter but often the video presentation as well!
This makes me wonder: imagine an algorithm to generate random sequence of odd numbers with a distribution similar to the primes. Might it be that for any sequence there would be a largest possible exception and beyond that all even numbers could be made from the sum of two numbers in the sequence. Call it the random Goldbach conjecture. Maybe the primes aren't particularly special (just lucky at low numbers) but it is the frequency that matters.
This also means that from 2105 onwards upto infinity, every natural number has a pair of equidistant twin primes, one to its left and one to its right.
Just noticed that for the Goldbach Conjecture involving Twin Primes, most of them are three consecutive even numbers that end in either 4, 6, or 8, so 10n+4, 10n+6, 10n+8. I wonder if there's a reason for that.
The obvious problem would be if twin primes were finite. Or even almost finite, like the gaps between twin primes get every large. Like with G greater than N.
it would be interesting to apply this to other prime pairs gaps, what about primes that are 4 apart? My conjecture is that primes any amount apart will have a finite number of exceptions, though the number of exceptions before you get to the golden land of enriched primes will be exponentially higher. if true that means there are ever restrictive but infinitely long list of prime numbers that can be goldbached if they are allowed a finite list of exceptions. also i noticed the exceptions for twin prime goldbach comes in triads, it seems like 0 would be the first of that triad. so maybe both 0 and 2 are actual exceptions to the original goldbach conjecture, perhaps that also has some meaning.
Isn't that strange that all the exceptions (apart from 2 and 4) go in triplets, almost all of them being (xx4,xx6,xx8) except one (400,402,404). Is there a reason for that ?
If I had a square. That square equated to 1 for area. I can make infinite squares inside. That is the 1 counting system. That is what you are using. You are telling me you are finding primes. They are all equal to the 1. It's the factors that are prime. The sides of the box or shape. 1 side x 1 side is 1. Only 1 is prime. Let's take 10. 2x5=10 10 is 1 because it's inside the box. 2 and 5 are outside. They are sides. They are primes. What to know more. What if I had 1x3. If I looked at the 1 from the side, all I would see is 1. If I looked from above, I would see 3. So direct sometimes matters with primes.. depending on what you are doing.
Say a set S of integers satisfies the 'Goldbach Property' if all integers greater than some threshold can be written as the sum of two integers from S. If we were to randomly populate an infinite set S with integers based on the frequency of the primes or twin primes, how likely is it that S would satisfy the Goldbach property?
If we assume that every even number can be written as the sum of two twin primes, could we show that the sum of the reciprocals of the twin primes must be infinite? (Of course we know that the sum is finite... Brun's theorem and constant.)
4:48 ... up to 20 billion? How long did that take, a minute? Not that it matters much how far up it's checked, it's just funny because usually, when you hear about how far up a conjecture has been checked, it's some giant number and you know it took some actual effort to do all that calculation. 20 billion is sort of just... tiny...
Okay, I checked them up to 200 billion. Took my desktop 40 minutes. Going higher would take a bit more work, because I just went with storing the full list of twin primes up to the cap and searching for the summing pairs in that, meaning I'm limited by memory. Anyway, to no-one's surprise, there were in fact no further exceptions in that range.
@@arirahikkala Dang you beat me to it lol, was writing my own code for this haha. Exact same thought, 20 billion is a pretty small range for something like this
@@arirahikkala there could be a distributed search for this. Once you've found a pair of twin primes for a given even number, nobody has to check it again.
7:35 I get two near misses above 24098, namely 24532 and 24536, which I think can only be formed from the twins (4091, 4093) and (20441, 20443). Can anyone tell me which partition I missed?
Not to nitpick, but if the list of exceptions for twin primes was only determined by trial up to like 20 billion, could there not still be infinitely many more exceptions? Like it doesn’t seem like the boundedness of the exceptions has been proven.
There definitely could. But given that the list of twin primes is growing, it's very unlikely that we will get another exception. Would be neat though.
@@oz_jones is the sheer size we are approaching actually a valid argument though? Considering scaling and whatnot is very often relative, how can we even make the argument that there is that big of a gap to the next one? I would argue we are potentially more being bounded by our imagination, although even then I’ll admit I’m not sure what meaning we’d derive from an equation for an infinite series of twin prime generation
Wow! It looks like there are always in a sequence, and all the sequences so far starts with 4. Bizarre. Sequences start small and looks like they get bigger (to max amount of 3)
Stopped the video at 5 minutes to type the following; It's odd to me that the exceptions are all sets of three (if you put 0 at the beginning), of the form XX0,XX2,XX4 or they're of the form XX4,XX6,XX8. There aren't any that aren't sets of three (provided you add 0), and there aren't any that start in one "ten" and end in another, like XX8,XX0,XX2 for example.
I'm assuming it has to do with the nature of them being twins. But I'm not smart enough to conjecture further. I'm sure minds brighter than mine have or are working on it.
I don't understand why 4 is an exception and it should not be possible to write it as the sum of two twin primes. James says it can be the same prime twice. So then 4=2+2 and 2 is twin with 3 as was said...
It's interesting how we can prove that there are infinitely many prime numbers by assuming that there is a largest prime number but we can't do that for twin primes.
Jane Street internships: jane-st.co/internship-numberphile
Extra footage from this interview: th-cam.com/video/d5IMSxRgeZk/w-d-xo.html
Hi
As a retired engineer I love these conjectures. Thanks
Sup, Beardy. Can you go ask the russian math guy over there (cant remember who, tell me, someone alex?? younger skinny dude), why he mocked ppl that Collatz Conjecture is a "commie PsüÖp to hinder western progress". As it is (but not about the progress, it´s just satanism) and it can be proved by anyone who has learned basic math (basically any 6th grader).
"I love a finite list" is basically saying "i love a condition that is SO PRECISE that it fits somewhere between zero and infinity"
He sounds like Liam Neeson.
I like transfinite lists. (Like fast growing hierarchies)
That is a surprisingly small space.
@@reidflemingworldstoughestm1394 Isn't it still an infinite amount of finite lists?
James Graeme hasn't changed since the first video I saw of him, just a little bit of white hair, but the same smile and cheerful spirit.❤
*Grime
@@Robi2009 *Prime
Graeme’s Fairie Tales?
Dude was born as anti-time.
*Dr. James Green...
I was waiting for a discussion of all those [n, n+2, n+4] triplets that kept coming as exceptions, but...
Same
Ah! Always first check for extra-footage, before writing a comment all-to-quickly... :-)
any sum of twin primes comes in an [n, n+2, n+4] triplet so exception sets make sense but without watching the extra footage you mentioned i dont know why the exceptions would always be triplets, and not quadruplets or something
Probably has something to do with the fact that a prime number n is a twin prime if n-2 or n+2 is also prime.
A pity 0 is excluded, the pattern would have included (0,2,4) at the beginning.
It's always great to see Dr. James Grime on the channel! He always teaches mathematics with a smile on his face!
More like Dr. James *Prime*
I'll see myself out.
@@oz_jones I appreciate a quality pun
@@piepiedog1this is a prime example of that
8:32 is absolute genius! I had not thought of that before Brady suggested it.
I know we (rightly) praise the professional mathematicians on this channel, but Brady is a spark of light in his own right, and deserves much more credit than he gets!
Indeed. Brady's questions and commentary add a lot to these videos.
James reaction was absolute gold.
I think you cannot proof the twin prime conjecture based on the twin prime Goldbach conjecture. Quite the contrary, I would think the twin prime Goldbach conjecture _assumes_ the twin prime conjecture is true. So I think the twin prime Goldbach conjecture would start with " _if_ the twin prime conjecture is true, …" or "Given infinite twin primes, …".
@@martinnyolt173 I don't see that as an assumption at all. It's a simple statement that is either true or false. Disproving the twin prime conjecture disproves the twin prime Goldbach conjecture.
This is what happens when you hang out with people who have more experience in a field (in this case, mathematics) than you do, but are willing to talk to and with you about what's going on rather than talk down to you.
Note: it is possible for both the twin prime conjecture and the Golbach conjecture to be true, but the Goldbach twin prime conjecture could be false. This is because the twin primes could get sparse too quick, that some massive number exists that isn't the sum of two of them. Think of all numbers that are the sum of two squares: There are infinitely many squares, and infinitely many sums, but there are also infinitely many integers that are *not* the sum of two squares. In fact, we already know in some sense twin primes do become sparse faster than primes: the sum of the inverses of twin primes is finite, but the sum of the inverses of primes diverges.
Correct. You would need to prove that there's a twin prime between it and twice the value. T
Conjecture: Every even prime can be written as the difference of two twin primes
Lol. So funny. And I feel so dorky for thinking that's funny.
Conjecture: Every even number greater than two can be written as a product of primes 😀
Yes, 2 is equal to 2.
@@Giannhs_Kwnstantellos wdym, a theorem is true; a conjecture isn't necessarily and has not been proven or disproven yet
@@GDominusOnYt I misread
On the flipside, there's the cautionary tale of the Mertens conjecture. Computational evidence is suggestive but not a proof!
Brady : there are infinitely many twin primes!
James : yeah!
OMG James proved the twin prime conjecture and isn't ready to publish yet
I had to do a double take when he said that
I have discovered a truly marvelous proof of this, which this video clip is too narrow to contain
"Maybe"
@@jackeea_ Me too, but I will leave it as a simple exercise to the viewers
9:56 👋 when we reconvene here in a few years because the proof did use this unlikely method, then we'll know my powers of Sight are working 🧠
The fact that even after many centuries of trying, no one in the wold has discovered neither an intuitive or convoluted proof to such a fundamental and seemingly trivial question is why I love mathematics.
Love the idea of a conjecture that proves the twin prime and goldbach conjecture at the same time lmao!
The real test is finding a non-trivial conjecture that simultaneously proves the Riemann hypothesis and 3n+1.
That would be such a human thing to happen.
"Solve these problems seperately. "
Humans: That's pretty tough...
"Solve these problems at the same time!"
Humans: Aight, bet
Dont know how i landed on this channel as i was never a numbers guy now cant get enough. Lmao! Im 44 now but not discouraged. Algebra 1 here we go...
Quite interesting. What I notice is other than 2 and 4, the remaining 33 exceptions are 11 'triples' of three consecutive even numbers each.
Check our extra footage at Numberphile2
If you include 0, then they're all in triples.
And all the triplet sets ending in 4,6,8.
@@madcapprof almost all! There's 400, 402, 404 - and what I find cool is that it is not the smallest triple, there's 94, 96, 98 before it.
always a pleasure watching one of your videos - especially anything about prime numbers!
It definitely feels like 0 should be included as an even number that can't be written as the sum of two twin primes, so that it could form a triple with 2 and 4
Amazing observation
I think it's the reverse: both 2 and 4 should be excluded. The reason for 2 is that the regular Goldbach states that every even number x >=4 can be written as the sum of two primes, so for Twin Goldbach we should also do x >= 4. And for 4, I would argue that 2 should also be on the list of Twin Primes. We could easily modify the definition and say that p1, p2 are Twin Primes if |p1 - p2| is *at most* 2, thus including 2, because |2 - 3| = 1
You took my comment!😊
-2+2
@@tomeklipinski4643 Negatives aren't prime, if my memory serves me right
3:43 That's some fresh out-of-the-box photography/editing
It reminded me of I think Sesame Street.
I'm looking forward to the extras.
If you are just taking even numbers, they can be described as sums of unique powers of 2 (or more like, every integer can, but even numbers don't have 2⁰). Essentially taking the binary representation of the number.
Summing two of them to get another number, is just summing more powers of two together, and removing that "uniqueness" as certain powers can be duplicates. You can easily reverse the process, take a binary number and do the steps backwards with literally any integer smaller than the target to get a second number you can sum that might or might not share powers of two.
If you're summing two primes together, they're also just like summing unique (minus the duplicates they share) powers of two. So it's instantly a lot more realistic that you could get any number from that. As to why it's only even numbers, both primes are odd by definition and as such both have 2⁰, meaning their sum will NOT have 2⁰.
The question then shifts: if you picked any even number, you could easily deconstruct it into two binary numbers. But if the conjecture is true, then there is ALWAYS a way to do so such that not just one but BOTH numbers are prime. Does this process always require shared powers? Because you can do so without shared powers (literally just split the target number in half high/low bits) but that doesn't guarantee the two halves will be primes. Could you even get two primes that don't share any powers (outside of 2⁰ which by default all primes must have)? Instinctively it feels like they need to share some, as some examples share all powers (namely ones where both primes are the same).
It also links to Mersenne primes and related primes, defined as 2ⁿ-1, meaning by definition they contain all powers of 2 up to a certain point: can you make any even number by restricting yourself to at least one of the two primes being a Mersenne prime? That would by definition require shared powers (again outside of the trivial 2⁰), unless the second prime is either 2ⁿ+1 (the twin of the first) or at least 4 times the Mersenne one.
Given the rarity of both Mersenne primes and primes in general, you'd need a lot of primes between the Mersenne ones, to be able to fill in for every even number, as such I don't think it's doable with this Mersenne limit.
Such an interesting topic.
I can hardly express how disappointed I am that the 4-6-8 sequence in the exceptions wasn't discussed.
Legends know that the wooden ladder against the wall takes us to all the greatest mathematicians of the past😂
This is a SUPER video. Thank you for this.
"I love a finite list" is only an interesting thing to say for a mathematician. Most of the rest of the world doesn't have any other option :)
“We’ll make start!” Love James
10 hours of Dr James Grime reading positive integers to fall asleep to
Digits of pi...
I suppose reading complex numbers would be more troublesome... lol
You might like watching darts ;)
For a good example of why checking a conjecture up to huge numbers isn't a proof of that conjecture, read about Mertens Conjecture. It's been shown to be false with the first counterexample lying somewhere above 10^16 but below 10^(4*10^28). Perhaps unsurprisingly, no one's found an explicit counterexample yet.
I love the idea of James looking at probably the two hardest problems in known math and going, "but what if both of them at the same time?"
1:36 there’s infinitely many of those
So controversial. So much money for that proof
01:34 "We think"
@5:00 Has this list of exceptions made it to OEIS yet??
I was thinking about primes yesterday, specifically prime bases, and wondering if James would be interested in doing a video about common properties of prime bases. I was thinking about prime bases on my walk home from work yesterday morning and I started wondering about whether there would be a set of properties that all such bases would have in common.
Bro straight up listed 35 numbers in real time. That's craaazy. They do be numbers
I'm not so intrigued by the fact that the exceptions come in triplets because I guess they're in "blind spots" that are just too far away from twin primes on both sides. But I'm fascinated by the fact that all of these triplets except one end in 4, 6, 8 and not 0, 2, 4 or 2, 4, 6 or 6, 8, 0 or 8, 0, 2. I'd love an explanation of that, and why there is one exception within the exceptions!
They should come back and shoot more footage. Maybe upload it to some secondary channel. And name it "extra footage", that would be fun.
They could even put a link to it into description ;)
Here's something I was thinking about:
Let's assume that Goldbacha conjecture ALMOST fails infinitely often, meaning that there exists large even numbers with only one way (up to order of summation) to write them down as sums of primes.
Now, given that n is almost a counter example, how difficult (what is the complexity in log(n)) is it to find primes p,q such that p+q = n?
Is it as exponentially difficult in log(n) as factoring semi-primes? Can one make an encryption method out of it?
didn't James say that the original Goldbach conjecture doesn't have "near misses" (numbers represented by only one pair of primes)?
@@vsm1456 We don't know that, if we did we already would have known the conjecture to be true.
One of the most number-filled numberphile videos of all time.
So, 404 Not Found?
This was hilarious😂
Thank you!
Have we checked to see how many exceptions there are for the versions for cousin primes, sexy primes, and other gapped pairs? I expect the number of exceptions grows. But I wonder if there’s a point where there are infinite exceptions.
that was a surprise. i will never forget that 94 cant be written as the sum of 2 twin primes
Do bananas sing of electric math?
404: twin primes not found
Now you need to get three blue one brown to set up an analog style visualization for it.
Twin prime conjecture is that their doesn't exist a natural number N such that all n greater than N are of forms 6ab+a+b,6ab-a-b, 6ab-a+b, or 6ab+a-b
Citation please?
If one includes the differences between two twin prime numbers, many of these entries will fall from the list. Example: 4204 = 86927 - 82723. [86927, 86929] and [82721, 82723] are twin prime pairs. This is the only pair of differences for 4204. A quick calculation pares the list to [94, 514, 516, 518, 904, 906, 908, 1114, 1116, 1118, 1264, 1266, 1268]. Maybe worthy of inclusion in OEIS.
Cool. 1,1 = 2 (10 base counting system(binary)) 1,1,2,2 = 6 ( 32 or 12 counting system (trinary)) 1,1,2,2,3,3=12 (60 or 12 counting system( quad). Binomial to monomial ? You termed them up, making them prime. 1,1 = 2 for 10. 1,1,2,2 =6 for 100. 1,1,2,2,3,3, = 12 for 1000. It can not be 28 unless the counting system is not synced with the others or a greater counting system is present. Its effecting the smaller ones. Thats mostly what you are seeing. I'm on god level.🎉
This is the video I've been waiting for!
I can't take my eyes off of the unsolved rubik's cube
The numbers that can't be made have the same relationship as sets of three even numbers that can be made by two pairs of twin primes. Every duo of pairs of tp add up like this: (sm=smaller pair bg=bigger pair)
sm1+bg1 sm1+bg2 sm2+bg1 sm2+bg2
But sm1+bg2 must be equal to sm2+bg1, so there are only three consecutive even numbers that are made by the two pairs. The ones that fall between the cracks have to be related in the same way, QED!
Find the general solution of the given differential equation
(1-x²)(1-y)dx = xy (1+y)dy
Is there a limit to the difference between the upper prime of a pair and the lower prime of the next pair? Or does the gap grow without bound?
The early primes have an unusual amount of twinning
One possibility is Goldbach’s Conjecture may be true but unprovable. It may just be one of those things which is true but we cannot know it to be true. Could this be proven?
New video idea: James Grime Says Numbers That Can't Be Written As The Sum Of Two Twin Primes For 1 Minute And 16 Seconds
The thing is, if the Goldbach conjecture (strong version shown in this video, not the twin prime or weak version though) is shown to being equivalent to an axiom of the system of math and numbers we use, wouldn't it then become necessary but impossible to "prove" since it would just be a result of the system we're using? I've always wondered if the Goldbach conjecture is impossible to prove (or disprove). The reason being this: What is the Goldbach conjecture? It isn't claiming that every even above a certain value can be described as the addition of two primes, it is actually claiming that we only need two primes. So, it *could* be described maybe with more than 2 primes, but the minimum amount needed is two primes. That's an important distinction because if we're trying to prove through contradiction we'd have to show the opposite.. so we'd have to show for at least one number value that it needs more than two primes added to be created. But, how would we prove that is or isn't the case unless we looked at every number and every amount of primes needed (so for example maybe some number way out there requires 6 or 8 primes to be formed. While I don't believe this to be the case, how do we exhaustively prove or disprove it? I don't think it is possible since we have to check infinite numbers at infinite varying amounts of primes needed to be added to form them. Does this *prove* that the Goldbach Conjecture cannot be proven? No. But, it does suggest it cannot be proven (even if it is indeed true). And, since I'd like to be the one to prove it (or at least be there when it is proven).. this possibility is a little depressing actually.
Great video though, I almost always love numberphile videos, especially their topics and subject matter but often the video presentation as well!
This makes me wonder: imagine an algorithm to generate random sequence of odd numbers with a distribution similar to the primes. Might it be that for any sequence there would be a largest possible exception and beyond that all even numbers could be made from the sum of two numbers in the sequence. Call it the random Goldbach conjecture. Maybe the primes aren't particularly special (just lucky at low numbers) but it is the frequency that matters.
4:31 Sold ! To the gentleman in the sixth row for 4208 style points
Have you watched Le Théorème de Marguerite?
This also means that from 2105 onwards upto infinity, every natural number has a pair of equidistant twin primes, one to its left and one to its right.
Just noticed that for the Goldbach Conjecture involving Twin Primes, most of them are three consecutive even numbers that end in either 4, 6, or 8, so 10n+4, 10n+6, 10n+8. I wonder if there's a reason for that.
do we know anything about the goldbach conjecture for primes separated by like 4 or 6? Or is it just twin primes atm
Good question!
My instinct is it will also apply but with longer list of early exceptions. But that’s just a guess.
The obvious problem would be if twin primes were finite. Or even almost finite, like the gaps between twin primes get every large. Like with G greater than N.
it would be interesting to apply this to other prime pairs gaps, what about primes that are 4 apart? My conjecture is that primes any amount apart will have a finite number of exceptions, though the number of exceptions before you get to the golden land of enriched primes will be exponentially higher. if true that means there are ever restrictive but infinitely long list of prime numbers that can be goldbached if they are allowed a finite list of exceptions. also i noticed the exceptions for twin prime goldbach comes in triads, it seems like 0 would be the first of that triad. so maybe both 0 and 2 are actual exceptions to the original goldbach conjecture, perhaps that also has some meaning.
The number of ways of writing even numbers as the sum of primes gets big quickly. 1,000,000 can be written 10804 ways and 10,000,000 in 77614 ways.
4:30 thats crazy that all the numbers are really small
3:44 I need a 1 hour version of this
Isn't that strange that all the exceptions (apart from 2 and 4) go in triplets, almost all of them being (xx4,xx6,xx8) except one (400,402,404). Is there a reason for that ?
0,2,4
All are triplets
Hi Dr. Grime! Hi Brady!
If I had a square. That square equated to 1 for area. I can make infinite squares inside. That is the 1 counting system. That is what you are using. You are telling me you are finding primes. They are all equal to the 1. It's the factors that are prime. The sides of the box or shape. 1 side x 1 side is 1. Only 1 is prime. Let's take 10. 2x5=10 10 is 1 because it's inside the box. 2 and 5 are outside. They are sides. They are primes. What to know more. What if I had 1x3. If I looked at the 1 from the side, all I would see is 1. If I looked from above, I would see 3. So direct sometimes matters with primes.. depending on what you are doing.
The longer it takes variable infinite quantium time never stops nother does knowledge
Say a set S of integers satisfies the 'Goldbach Property' if all integers greater than some threshold can be written as the sum of two integers from S. If we were to randomly populate an infinite set S with integers based on the frequency of the primes or twin primes, how likely is it that S would satisfy the Goldbach property?
If we assume that every even number can be written as the sum of two twin primes, could we show that the sum of the reciprocals of the twin primes must be infinite? (Of course we know that the sum is finite... Brun's theorem and constant.)
Interesting video
my brain is dead after that video so much information for a 9 year old
210 is very Goldbachy
Towards the end of the exceptions list got in a real "55 burgers, 55 fries, 55 tacos, 55 pies, 55 cokes, 100 tater tots..." cadence
Do the exceptions exist if you allow for the difference of two twin primes?
4:48 ... up to 20 billion? How long did that take, a minute?
Not that it matters much how far up it's checked, it's just funny because usually, when you hear about how far up a conjecture has been checked, it's some giant number and you know it took some actual effort to do all that calculation. 20 billion is sort of just... tiny...
Okay, I checked them up to 200 billion. Took my desktop 40 minutes. Going higher would take a bit more work, because I just went with storing the full list of twin primes up to the cap and searching for the summing pairs in that, meaning I'm limited by memory.
Anyway, to no-one's surprise, there were in fact no further exceptions in that range.
@@arirahikkala Dang you beat me to it lol, was writing my own code for this haha. Exact same thought, 20 billion is a pretty small range for something like this
My guess is that this was just something Matt Parker checked. Or Dr. Grime is using the long count version, so 20 million millions.
@@arirahikkala there could be a distributed search for this. Once you've found a pair of twin primes for a given even number, nobody has to check it again.
Can we compare the prime and the grime counting function?
7:35 I get two near misses above 24098, namely 24532 and 24536, which I think can only be formed from the twins (4091, 4093) and (20441, 20443). Can anyone tell me which partition I missed?
Not to nitpick, but if the list of exceptions for twin primes was only determined by trial up to like 20 billion, could there not still be infinitely many more exceptions? Like it doesn’t seem like the boundedness of the exceptions has been proven.
There definitely could. But given that the list of twin primes is growing, it's very unlikely that we will get another exception. Would be neat though.
@@oz_jones is the sheer size we are approaching actually a valid argument though? Considering scaling and whatnot is very often relative, how can we even make the argument that there is that big of a gap to the next one? I would argue we are potentially more being bounded by our imagination, although even then I’ll admit I’m not sure what meaning we’d derive from an equation for an infinite series of twin prime generation
James - Why the “sets” of three exceptions?
See the extra footage on Numberphile2
Wow! It looks like there are always in a sequence, and all the sequences so far starts with 4. Bizarre. Sequences start small and looks like they get bigger (to max amount of 3)
I might be able to disprove the Goldbach conjecture for twin primes.
I wonder why it many of those exceptions come in triplets of 10n+4, 10n+6, 10n+8.
Stopped the video at 5 minutes to type the following;
It's odd to me that the exceptions are all sets of three (if you put 0 at the beginning), of the form XX0,XX2,XX4 or they're of the form XX4,XX6,XX8. There aren't any that aren't sets of three (provided you add 0), and there aren't any that start in one "ten" and end in another, like XX8,XX0,XX2 for example.
what are your favorite finite lists!!!
What is 0° celcius plus 0°celcius in fahrenheit?
My favorite twin primes are 41 and 43.
3:21 I will now tell everyone that this is how all Numberphile videos look like. :P
What is known about thinnest subsets of the primes that satisfy their Goldbach variant(s)?
Brady Haran and James Grime, name a more iconic duo
weird how they're all triplets, and almost all of them end with 4, 6, or 8. wonder why that is
I'm assuming it has to do with the nature of them being twins. But I'm not smart enough to conjecture further. I'm sure minds brighter than mine have or are working on it.
3:15 Now THIS is a singing banana!
Primes are separated by dimensions?
I've been curious about this since a math teacher in manila claimed to prove this. weirdly he's keeping it secret so i don't know what became of it
I don't understand why 4 is an exception and it should not be possible to write it as the sum of two twin primes. James says it can be the same prime twice. So then 4=2+2 and 2 is twin with 3 as was said...
2 and 3 arent concidered twins because the gap is 1 instead of 2
@@oz_jones Ah, yes! Of course you're right..!
It's interesting how we can prove that there are infinitely many prime numbers by assuming that there is a largest prime number but we can't do that for twin primes.
we need a Numberphile Enigma Code remastered version
3:30 404 error: twin prime not found!
Does the twin prime conjecture only work in base 10?
Neither the twin prime conjecture nor the Goldbach conjecture depend on the base, since none of the concepts involved does.