What's the next freak identity? A new deep connection with Sophie Germain primes

0:06 Welcome to another Mathologer video.

0:08 You all know those two freaky identities over there, right?

0:11 And for most people they are just that: pretty

0:14 freaks without any deeper meaning… Unless of course,

0:16 you are a Mathologer regular.

0:18 Because if you are, you would have found out

0:20 in two earlier videos that these identities and their underlying equations,

0:25 those two there lead very interesting secret lives.

0:29 Heron’s formula, the heptagon golden ratio,

0:32 and all that jazz.But what even most Mathologer

0:33 regulars won’t know yet is that these two identities up there are just the start

0:34 of a very interesting infinite sequence of similar identities.

0:34 Really?

0:34 Yes, and in fact, there is

0:34 a whole world of sum-equals-product monsters out there.

0:34 Let’s go and explore :)Okay, infinite sequence,

0:35 hmm, so what’s next (point) over there?

0:35 Hmm.

0:35 Well, obviously, the next equation at the bottom

0:35 is this one here.If you’ve not done so already,

0:36 definitely check out those two earlier videos

0:39 for a lot of amazing and beautiful maths :)Anyway,

0:42 what even most Mathologer regulars won’t know

0:44 yet is that these two identities up there are just the start of a very

0:49 interesting infinite sequence of similar identities.

0:52 Really?

0:52 Yes, and in fact, there is

0:55 a whole world of sum-equals-product monsters out there.

0:59 Let’s go and explore :)Okay, infinite sequence, so what’s next?

1:03 Hmm.

1:03 Well, obviously, the next equation at the bottom is this one here.x plus y plus

1:09 z plus u equals x times y times z times u:) And, at the top,

1:16 we are looking for a solution of this new equation in positive integers.

1:20 So, to start with, we want four positive integers on both

1:23 sides of the equal sign at the top.Can we spot a pattern?

1:27 Well, if we shuffle things around like

1:29 this, as I did in the thumbnail of this video,

1:32 things are starting to look very SUM-Metrical.

1:35 Okay, okay, bad one :)2, 3.

1:37 What comes next?

1:39 4, of course :)Then2, 2, what’s next?

1:42 Another 2, maybe?Now.

1:44 1.

1:44 What’s next?

1:45 Well, let’s first add and multiply what we’ve got so far.

1:50 2+4 that’s 64 times 2 that’s 8.

1:52 Now, if you’d have to bet your life on what number comes after 1 in this set-up,

1:58 you’d probably go for 1, right?

2:00 Okay, so let’s try 1.

2:02 On the right, multiplying by 1the product actually does not change.

2:07 Aha!

2:07 Can you see what’s happening here?

2:09 On the other hand, adding the corresponding 1 on the left ups the sum by 1.

2:14 7 and 8, almost equal.

2:16 Obvious how to finish, right?

2:18 Just chuck in another 1 :)1 on the right.

2:21 Nothing changes.Add 1 on the leftTada:) Equality.

2:25 What’s next?

2:26 Well, After 4 comes 5.

2:29 So replace 4 by 5.

2:30 That makes the sum on the left go up by 1.On the right,

2:34 replacing 4 by 5 ups the product by 2.And now we know what to do.

2:40 Just pad with another 1:)And it’s clear sailing from here,

2:44 forever and ever after.2, 3, 4, 5, 6, and so on, in the middle.

2:49 Then2, 2, 2, 2, 2 and so on.

2:52 And pad out the rest with 1s such that in every identity the number of terms

2:57 on the left and right (click) is equal to the largest number in the middle.

3:02 There (point) three terms and 3 in the middle.4

3:05 terms and 4 in the middle, and so on.

3:09 Two remarks: One: The whole padding with 1s business may come across a little

3:14 bit like cheating but if you want to play this game with positive integers,

3:18 you really don’t have a choice.

3:20 Because, if you also insist on not involving any 1s,

3:23 then 2+2=2x2 is the only identity that works.

3:28 Second remark: Taking this padding-with-1s-idea one step further,

3:32 it’s clear that there are many more of these sum-equals-product identities.

3:36 Just start with a random assortment of positive integers, say 2 5 7.

3:42 Doesn’t matter which, you can even have some numbers repeat.

3:46 Like for example start with the numbers

3:48 2 5 5 7.Calculate their sum and product.Then,

3:52 as in this example, in general, the product will be greater than the sum.

3:58 Subtract the sum from the product.Then if you pad out the left and right

4:04 sides with that difference number of 1s you get one of our identities.

4:08 So, in this case add 331 1s on the left and multiply

4:12 the same number of 1s on the right and you get equality.

4:16 Super easy.

4:17 In fact, the first time I stumbled across all this I thought:

4:21 Okay, cute but in the end pretty trivial, right?

4:23 But let’s slow down:) After all, even if something is easy to understand,

4:27 it may still lead to deep and beautiful mathematics.Right?

4:30 You can explain to a 10-year old what a prime number is.

4:34 But at the same time some of the greatest mathematical theorems,

4:37 and some of the most important and hardest problems are about prime numbers.

4:43 As the Mount Everest example,

4:45 the Riemann hypothesis is all about primes :)And, it turns out that the world

4:52 of sum-equals-product identities is also full

4:55 of pretty surprising prime number marvels and mysteries.

4:59 Maybe not the Riemann hypothesis,

5:00 but still great stuff:) In the rest of this video,

5:04 I’d like to tell you about one

5:06 particularly nice mathematical marvel involving the mysterious Sophie

5:10 Germain primes.These were discovered 200 years ago

5:13 by– you guessed it:)– the mathematician Sophie Germain.

5:17 She discovered these primes while working on Fermat’s last theorem one

5:21 of the most famous open mathematical problems

5:23 at the time.But before I get started,

5:26 I’d like to take this opportunity to pick all

5:29 your brains to dig for mathematical treasure :)In your mathematical journeys,

5:35 did you ever encounter any of these sum-equals-product

5:39 identities or equations in surprising settings or contexts?

5:43 For example, my friend Marty recently told me about this paper by Stanford

5:48 mathematician Brian White in Inventiones Mathematicae,one

5:51 of the most prestigious maths journals.

5:54 Turns out that amidst a lot of very high-powered mathematics,

5:59 2+2=2x2 ends up being the ingredient that makes everything come together.

6:05 Really quite wonderful.

6:07 Anyway, if you know of anything like this for any of the sum-equals-product

6:12 identities and equations please let the rest of us know in the comments.

6:17 Okay, now let’s have a close look

6:23 at the world of sum-equals-product identities.Let’s call

6:33 the number of terms on one side

6:35 of a sum-equals-product identity the length of the identity.

6:39 So, 2+2=2x2 has length 2 and 1+2+3=1x2x3 has length 3.

6:46 It’s then natural to ask how many

6:50 sum-equals-product identities there are of a given length.

6:53 Well, it’s natural for us mathematicians to ask this:) It’s what we do:) So,

6:58 here straightaway is a plot of these numbers of identities up to length

7:02 10,000 taken from a really nice

7:05 paper by the Polish mathematician Zah-kar-CHEM-nee

7:07 that I link to in the description of this video.What a weird looking

7:12 plot:) Definitely not the nice curve that you may be expecting to see here,

7:17 but a standard plot regardless, consisting of one dot each for the possible

7:22 lengths between the smallest length 2 and length 10,000.

7:26 Right, take any length..then the height of the dot

7:29 above it is the number of sum-equals-product

7:32 identities of that length.in this case around

7:37 25.Clearly the dots are all over the place.

7:40 Meaning that the number of sum-equals-product identities

7:42 for different lengths are all over the place.

7:44 The general trend is a logarithmic growth.

7:47 But the trend is not very trendy, is it?As we increase the length,

7:51 we come across larger and larger numbers

7:53 of identities (click) but, at the same time,

7:56 really small numbers keep popping up.

7:58 (click) Let’s have a closer look at what happens for the smallest lengths.

8:03 Alright, we know that for every length there

8:06 is one of our basic sum-equals-product identities over there.

8:09 In fact, for length 2 there is only this one sum-equals-product identity.

8:15 That is, the equation X+ Y= Y x X only has one solution in positive integers.

8:21 The same is true for the next equation.The only solution (up

8:25 to shuffling of the terms) corresponds

8:28 to our second basic sum-equals-product identity.

8:31 Same for length 4.Then 5.But now it turns out that for length 5 we

8:40 also have this second sum-equals-product identity

8:43 and also this one hereBut then that’s it.

8:47 For length 5 there is a total of three identities,

8:52 those three (point)And now that we’ve got going,

8:55 there should be lots of identities of length 6, right?

8:58 Nope.

8:59 For length 6 there is just one again.And,

9:02 after this, things continue up to length 9

9:07 as follows.Here are two little challenges for you.

9:12 First, can you find all the identities of length 10?

9:16 Also, can you find a length for which there are more than 10 identities?

9:22 Leave your answers in the comments :)Okay,

9:25 so what’s a natural question to ask here?

9:28 Who cares?

9:29 No, no, no, no.

9:30 Natural or not, that question is taboo around here:) Seriously,

9:34 what might we ask about these lengths?

9:36 Well, how about “For which lengths

9:39 is there only one basic sum-equals-product identity,

9:42 the basic identity is the only one?” Right?

9:46 Natural question?

9:47 Well, so far we’ve seen four such lengths2, 3, 4, and 6.

9:52 Are there more?Yep, 24.It turns out there is

9:56 also only the one sum-equals-product identity of length 24.

10:02 And then, we also have 114174and 444What’s next?

10:07 Should I leave you to hunt for the next one?

10:12 Maybe not:) In fact,

10:14 mathematicians have used computers to check all lengths up to 10

10:17 billion and have not discovered another length with only the basic solution.

10:22 So, maybe that’s it, and the list ends with 444.

10:26 But we don't know.

10:28 Checking all the way to 10 billion is very suggestive,

10:32 but ten billion is still a long long

10:35 way from infinity :)How could we settle this question?

10:38 Well, we could leave the computer running:)

10:41 But that will take an infinite amount of time,

10:44 and we’re probably a little too impatient for that.

10:47 So, instead, how might we think our way to a solution?Well,

10:50 a good starting point would be to figure

10:53 out what all these special lengths have in common.

10:57 Right?

10:57 Okay, so have a look at those numbers up there.

11:00 Is there any pattern, any common trait that sticks out to you?

11:05 Hmm.

11:06 Lots of 4s?

11:07 Yes, but sadly that’s not it :)Now, I could just tell you the surprising answer.

11:13 But that’s not the Mathologer way:) Instead, I will do what I often do,

11:18 I’ll kidnap you, and take you on a journey of discovery,

11:21 which leads naturally to this answer.

11:23 Let me know in the comments whether you enjoyed being kidnapped.

11:35 Okay, let’s start with something easy.

11:38 Let’s figure out why, for length 2,

11:40 there is only one positive integer solution of this equation, namely 2+2=2x2.

11:46 How would you get started on this mission?

11:51 Easy!

11:51 Solve for one of the variables,

11:53 say Y and hope for the best:) Algebra autopilot on.Okay,

12:00 just one Y left and it would now be easy to finish solving for Y.

12:11 However, at this point an experienced

12:14 algebra torturer may notice something else,

12:16 an opportunity to simultaneously isolate both variables:) Have a look.

12:21 Take minus 1 on the rightand add 1 to cancel out the -1.

12:26 Right, -1+1, cancels out to 0, so nothing’s changed.

12:30 Except…Can you see it?

12:32 ThereThat’s nice isn't it?

12:36 We managed to rewrite our initial equationin a second supersymmetric way.

12:45 In particular, this means that both equations have exactly the same solutions.

12:50 And, so, what are those solutions?

12:52 Well, easy.

12:52 Since X and Y are supposed to be integers,

12:55 so are the numbers X-1 and Y-1.But then what this says is

13:00 that the product of the orange integer and the green integer equals 1.

13:05 Of course this means that either both orange and green

13:08 are equal to 1 or both are equal to -1.

13:12 Right?

13:12 1 times 1 is 1 and -1 times -1 is also 1, and that’s all.

13:18 Now, if both numbers are equal to 1.Then, well, what?

13:23 X-1 is 1.

13:25 So X=2 and so is Y.That’s our 2+2=2x2 solution.

13:31 On the other hand, if both orange and green are -1then we have X-1=

13:39 -1 and so X= 0 and so is Y.0+0=0x0 that’s also correct but we decided,

13:48 well rather I made an executive decision for all of us:)

13:54 that we’d only be interested in solutions with positive integers.

13:58 Alright, that was definitely a slick way to show

14:01 that we have only one solution for length 2.

14:04 But, more importantly, all this mucking around leaves us just

14:08 one step short of another crucial insight.Okay,

14:12 again, this nicely symmetric equation is equivalent

14:16 to our initial sum-equals-product equation.But here’s another trick.

14:20 Just shuffling everything in the bottom equation to the left of the equal sign,

14:24 we obtain a third equivalent equation.Does

14:27 the quantity on there left ring a bell?

14:30 That difference between a product and a sum?Right,

14:33 if you start with two positive integers X and Y,

14:37 then this difference tells you how many 1s you need

14:40 to pad these two numbers with to get one of our identities.

14:44 In the case of X and Y both being equal to 2,

14:49 this difference is 0.But now let’s consider any old X and Y.

14:53 Then the difference between the product and the sum is some other number,

14:57 let’s call it D.Then the exact same algebraic acrobatics as earlier give us

15:03 this equivalent equation at the top.Easy for you to nut out the details.

15:07 And of course you should.

15:09 No freeloading here:) Anyway, let’s check using an example:

15:15 X=2 and Y=7.What’s that difference?2 times 7, that’s 14 minus 2+7,…, 9,

15:22 14-9 is 5.Okay that means we need to pad 2 and 7 with five 1s.

15:29 And, actually, if we do that we just get one of our basic identities.

15:36 That one here.Now what about at the top?X-1, that’s 2-1.

15:42 2-1 is 1.andY-1, that’s 7-1 that’s 61 times 6 is 6Works.

15:49 That 6 is the difference 5 from before plus 1, 5+1 is 6.

15:55 Now here comes the nifty bit.

15:57 6 is also 2 times 3.But, then, if X-1 is 2then X is 3.And if Y-1 is 3,

16:06 then Y is 4.But, now we can use these new X and Y,

16:12 to get another difference of 5 at the bottom.

16:17 3 times 4 is 12 minus 3+4, 7,

16:22 12-7 equals 5.But now that means that if we pad out 3 and 4 with five 1s,

16:29 we obtain a second sum-equals-product identity of the exactly

16:33 the same length as the basic identity we started with.

16:36 Can you see where I am going with this?

16:39 Well, what we just discovered boils down to a nice

16:42 way of creating new sum-equals-product identities from our basic ones,

16:46 of the same lengths.

16:48 Let me spell out this streamlined way of creating

16:53 new sum-equals-product identities using a couple of examples.

16:56 Okay?

16:56 Well, let’s start with the basic

16:59 sum-equals-product identity that we just encountered,

17:01 2 and 7 padded out with 5 1sAlright, 7 minus 1,

17:07 that’s the 6.6 is 2 times 32 plus 1 is 3 and 3 plus 1 is

17:16 4And that’s our new sum-equals-product identity of length

17:20 7 once again.Now let’s start with the basic identity

17:25 of length 5.5 minus 1 is 44 is 2 times 22 plus 1 is 3And there is

17:35 the new sum-equals-product identity of the same length

17:39 as the basic one we started with.Does this always work?

17:43 Well, here is what quite a few of you will already be

17:48 waiting for by now.6 minus 1 is 5But now, since 5 is prime,

17:54 the only way to write 5 as a product of positive integers

17:59 is 1 times 5Adding 1 to both 1 and 5, gives…2 and 6.

18:05 Which after padding with 1s only gets us back to where we started.

18:09 In general, for a given length N our construction will

18:15 give no new sum-equals-product identities if N-1 is a prime number

18:20 and at least one new sum-equals-product identity if N-1 is not

18:25 a prime number.And that tells us something very interesting about our puzzle.

18:30 Remember, our puzzle was to try to determine,

18:33 without using a computer for infinity hours, for which special lengths N there

18:39 exists just the one basic sum-equals-product identity.

18:42 And this is what we just discovered:

18:45 any of those special numbers minus 1 must be a prime number.

18:49 How neat is that?

18:51 Let’s check.

18:52 Those are the known special lengthsAll minus 1.1, 2, 3, 5, 23, etc.

18:57 Works, those are all prime numbers.

19:00 Well, that 1, the first of the numbers,

19:02 that’s not a prime, but close enough, right?

19:04 1 used to be a prime and is at least still

19:07 an honorary prime:) Nothing goes into it except 1 and itself,

19:11 which is what matters here.

19:13 Anyway, what we’ve discovered is that if

19:15 there is another one of these special lengths, does not matter how large,

19:20 then that special length minus 1 must be a prime number.So,

19:24 we only need to look for special lengths among

19:27 those numbers that diminished by 1 are prime numbers.

19:30 Hmm, that’s all definitely some pretty prime number action here:) Okay,

19:35 but still there are a lot of prime numbers, and so there must be something extra

19:40 special about those particular primes up there.

19:43 To track down that extra speciality:) let’s have

19:47 another close look at our line-up of small cases.

19:53 All clear so far?

19:57 Great?

19:57 Again, over there is our list of small cases.

20:04 What I’ve hightlighted in this list are

20:06 the first three in the infinite family of special

20:09 identities that our streamlined way of making

20:12 new sum-equals-product identities from the basic ones produces.

20:15 Remember, this infinite family and the streamlined way

20:19 of producing it resulted in our neat prime number insight.

20:23 Another close look at our list of small cases,

20:27 yields a second infinite family of identities.

20:29 This second infinite family and the rule underlying it will lead

20:33 us to another really neat prime insight about our special lengths.

20:38 Ready for this?

20:40 Okay, let’s go.Alright,

20:41 here are the first two identities in our second family.Hmm…What’s the rule here?

20:47 How do you go from length 5 at the top

20:51 to 2,2,2 in the yellow box or from length 8 to 2,

20:54 2, 3 in the blue box?Well we’ve not got much to work with so far.

20:59 So let me also show you the next

21:02 couple of identities in this family.222, 223, 224.

21:05 So we’re all guessing that 225 is next, right?

21:10 Well, let’s see…233 not 225:) In fact, if you look at a couple more examples,

21:17 the only obvious pattern is the fact that we always get a 2 and something,

21:23 and something else.But, since that one 2 is common to all these new identities,

21:28 maybe the better question to ask is how does the 5turn into 2 2.

21:33 And how does the 8turn into 2 3?

21:35 Well, actually, it’s not hard to figure out what the general rule is,

21:40 now that we know what we are gunning for :)Right, starting with some N,

21:45 we want to find X and Y up

21:48 there such that both sum-equals-product identities have length N.

21:52 And how do we do that?Easy, do some algebra torturing:) First,

21:58 the number of 1s in the basic sum-equals-product identity at the top is what?

22:05 Well, the length is N and only the 2 and the N are not 1s.

22:10 And so there are N-2 1s.

22:12 Right?On the other hand,

22:14 the number of 1s in the second sum-equals-product identity is,

22:19 as usual, the product of 2, X and Y minus the sum of 2, X and Y.Next insight:

22:26 There is one more 1 in the first identity than in the second.

22:29 Have a look.One more 1 at the top.

22:31 So adding 1 to the second expression

22:33 makes both expressions equal.Now simplifying on autopilot

22:37 and isolating the X and Y as previously

22:42 recasts this equation in this supersymmetric form.

22:45 At this stage the algebra involved should be a piece

22:51 of cake for you and so I won’t bother

22:56 with the details.For comparison the corresponding equation for our first

23:01 rule is this.So exactly the same except for the 2s.

23:05 Now, remember, the first equation translates

23:08 into this rule for generating new special sum-equals-product identities:

23:12 subtract 1 from NWrite N-1 as a product.Add 1

23:16 to both factors to get the new X and Y.

23:21 Similarly, for our new second rule we multiply N

23:26 by 2 and minus 1Write 2N-1 as a productAnd,

23:31 finally, add 1 to each factor and divide by 2 to get the new X and Y.

23:36 Right?

23:37 And here is the new rule in practice:2 times N-1,

23:42 so 2 times 5 is 10 minus 1 is 9write as a product,

23:48 9 is 3 times 3add 1 to each factor and then divide by 2.

23:54 3 plus 1 is 4, divided by 2 is 2.throw in that common 2.E voila,

24:00 here is our new special sum-equals-product identity,

24:03 exactly as long as the basic one we started with.Another example2 times 6 is 12,

24:10 minus 1 is 11A prime number and we are stuck.

24:16 Next,2 times 7 is 14, minus 1 is 13stuck again.2 times 8 is 16 minus 1 is 15aha,

24:26 3 times 5+1 divided by 2, both of them.

24:29 3+1 is 4 divided by 2 is 2 and 5 plus 1 is

24:33 6 divided by 2 is 3… 2 3join in the common 2 in front.

24:39 And done.

24:40 :)Okay, where does this leave us

24:42 in terms of characterising our special lengths?Well,

24:46 we already know that if N is a special length, then N minus 1 must be prime.

24:53 But now with our new rule, we also know that, in addition,

24:58 twice N minus 1 is prime as well.

25:01 Right?

25:01 There, all those new numbers 3, 5, 7, 11, 47 etc.

25:06 are also prime.

25:07 And, so, if we ever find another one of those special lengths,

25:11 just times it by 2 and minus 1, and you get another prime.

25:15 Super interesting and surprising, don’t you think?

25:19 Well, it sounds unlikely, doesn't it?

25:21 What are the chances that both N-1 and 2N-1 are prime numbers?

25:27 Or, phrased differently, what are the chances that, given a prime number P,

25:32 twice P plus 1 is also a prime number?As

25:35 I already mentioned at the beginning of this video,

25:38 it was the mathematician Sophie Germain (click) who was

25:41 the first to ponder this question around 1800 while

25:44 trying to prove Fermat’s last theorem.Because of this, a prime

25:47 number P such that 2P+1 is also a prime

25:51 is now called a Sophie Germain prime.One of Sophie

25:54 Germain’s major claim to fame is that using her primes

25:58 she was the first to make significant progress towards

26:02 proving Fermat’s last theorem for infinitely many prime exponents.

26:05 Prior to her research people only considered individual exponents.

26:10 If you have not heard of Sophie Germain,

26:13 definitely check out the remarkable story of her mathematical life.

26:17 Completely self-taught,

26:17 had to pretend that she was a man to enter maths competitions,

26:21 the whole bit.Okay, history lesson done.

26:24 Back to work.

26:25 So we know that every special length minus 1 is a Sophie Germain prime,

26:32 right?What about the other way around?

26:35 Is every Sophie Germain prime also a special length minus 1?

26:40 The answer to this question is… Sadly, or not sadly,

26:44 depending what you are cheering for, there are

26:46 a lot more Sophie Germain primes:) As it happens,

26:49 Sophie Germain primes are nowhere near as rare as one might expect.

26:53 To illustrate, here are all the Sophie

26:55 Germain primes up to 113.Computer searches have

26:58 produced ridiculously large Sophie Germain primes which

27:02 also happen to be useful in cryptography.

27:04 In fact, the people in the know believe that there may well be

27:08 infinitely many Sophie Germain primes.Apart from the two

27:12 special infinite families of sum-equals product

27:15 identities that led to our prime number insights there are of course many other

27:21 infinitely families one can investigate and milk

27:24 for further insights into our special lengths.

27:26 This is in fact possible, however nothing quite as neat as what

27:29 I’ve talked about today has been discovered.

27:32 Anyway, check out the linked papers

27:34 in the description of this video for more details.Ooookay,

27:38 what other interesting open sum-equals-product results and questions are there?

27:43 Let’s finish up with that.First off, if I had to bet my life whether or not

27:56 those numbers up there are the only special lengths,

28:00 with just one sum-equals-product identity each, I’d go for “Yes”,

28:04 those are the only ones:) But, then, what about the lengths that correspond

28:08 to exactly two special sum-equals-product identities?

28:10 A couple of the small lengths that we considered earlier were of that type.

28:16 There, at the bottom, there are exactly two identities each for lengths 7,

28:20 8 and 9.Turns out that the list of known

28:23 lengths of this type is also fairly short.

28:26 To be precise we know of 49 such lengths,

28:31 the largest known being 6324There are also some primes hiding in these numbers.

28:38 Turns out that if there are exactly two special identities of length N,

28:44 then the number N-1 or the number 2N-1 or both are prime.

28:49 For the first 13 in this list (click) here are the corresponding N-1s

28:55 and 2N-1s.And here are the primes among

28:59 these N-1s and 2N-1s.That’s a pretty cute result, too, don’t you think?

29:05 Now it’s possible to show that that even if

29:07 one of these N-1s or 2N-1s is not a prime,

29:11 such a non-prime is nevertheless always close

29:14 to being a prime number in a certain sense.

29:17 What does it mean to be close to being a prime?

29:20 To find out, check out the details in the description

29:23 of this video.By now you will probably not be surprised

29:27 to hear that all sum-equals-product identity experts also suspect that those 49

29:33 lengths are the only ones corresponding to exactly 2 identities.

29:38 In fact, it’s reasonable to conjecture that for any fixed number

29:43 of identities there are only finitely many lengths corresponding to that number.

29:47 All very intriguing and lots of open interesting

29:50 problems remain to be tackled in this area.

29:53 Anyway, that’s it for today.

29:55 Don’t forget to let the rest of us know

29:58 of any interesting occurrences of our special identities in the wild.

30:03 Picking the brains of zillions of people in this respect was

30:07 one of the main reasons for me to put together this video:)