Advice
We advocate a simple minded AI approach to pure maths research: start with a basic, central object in mathematics, and just systematically explore the possibilities adjacent to it, where adjacent means roughly that we perform small variants and see what happens.
To illustrate the strategy, let's start with one of mathematics' most important objects: the exponential polyseries (no! it is not a "function"!!) and then let's move in the same direction as we did to get the Bernoulli numbers and the Faulhaber polynomials for sums of powers of natural numbers. And lo and behold! Surprises await.
There is enough potential exploration in this lecture to keep an amateur mathematical investigator busy for months and months. That could be you!
Here are the Wild Egg Maths Playlists (some available only to Members!)
Algebraic Calculus One (for the Open Learning Course) : https://www.youtube.com/playlist?list...
Algebraic Calculus Two (for the Open Learning Course in preparation): https://www.youtube.com/playlist?list...
Advice for Mathematics Exploration (for Members only): https://www.youtube.com/playlist?list...
Solving Polynomial Equations (Members only): https://www.youtube.com/playlist?list...
Hexagrammum Mysticum (Members only): https://www.youtube.com/playlist?list...
Algebraic Calculus and de Casteljau Bezier curves (Members only): https://www.youtube.com/playlist?list...
Exceptional Structures in Maths and Physics via Dynamics on Graphs (Members only): https://www.youtube.com/playlist?list...
Maxel Inverses and Orthogonal Polynomials (for Members only): https://www.youtube.com/playlist?list...
Triangle Centres: https://www.youtube.com/watch?v=iLBGX...
Six: An elementary course in pure mathematics: https://www.youtube.com/playlist?list...
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Here are the Insights into Mathematics Playlists:
Maths and Music: https://www.youtube.com/playlist?list...
Diffusion Symmetry: https://www.youtube.com/playlist?list...
Elementary Mathematics (K-6) Explained: https://www.youtube.com/playlist?
list=PL8403C2F0C89B1333
Playing Go: https://www.youtube.com/playlist?list...
Year 9 Maths: https://www.youtube.com/playlist?list...
Ancient Mathematics: https://www.youtube.com/playlist?list...
Wild West Banking: https://www.youtube.com/playlist?list...
Sociology and Pure Mathematics: https://www.youtube.com/playlist?list...
Old Babylonian Mathematics (with Daniel Mansfield): https://www.youtube.com/playlist?
list=PLIljB45xT85CdeBmQZ2QiCEnPQn5KQ6ov
Math History: https://www.youtube.com/playlist?list...
Wild Trig: Intro to Rational Trigonometry: https://www.youtube.com/playlist?list...
MathFoundations: https://www.youtube.com/playlist?list...
Wild Linear Algebra: https://www.youtube.com/playlist?list...
Famous Math Problems: https://www.youtube.com/playlist?list...
Probability and Statistics: An Introduction: https://www.youtube.com/playlist?list...
Boole's Logic and Circuit Analysis: https://www.youtube.com/playlist?list...
Universal Hyperbolic Geometry: https://www.youtube.com/playlist?list...
Differential Geometry: https://www.youtube.com/playlist?list...
Algebraic Topology: https://www.youtube.com/playlist?list...
Math Seminars: https://www.youtube.com/playlist?list...
0:01
Hello everyone i'm Norman Wildberger thanks for joining me on this adventure
0:04
i'm looking at Bernoulli numbers uh the last lecture and today i want to just give a little
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bit of a variant on what i was talking about and um and sort of talk a little bit about possible
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ai approach to mass research now in my opinion ai is going to be playing more and more of an
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important role in investigations in pure mathematics in the coming years
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and i wouldn't be surprised if in a few decades the ai basically dominated
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mathematics research in pure mass and that's a very significant development that we should
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all be a little bit nervous about but also perhaps excited and so today i want to just talk
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about you know how would you maybe orient yourself towards an ai approach what what would be involved
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so there's all kinds of sophisticated things that one can do but i'm just going to advocate this
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rather a simple idea that what an ai could do is to take some existing theory or object maybe
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which is relatively simply defined but uh sort of recognized as being of central importance and then
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just modifying it so just just investigating the space around it without any particular
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orientation in terms of motivation or trying to solve some problem or trying to create a theory
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just rather with the idea of surveying the possibilities adjacent to something that's
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interesting okay that's the basic idea here's something interesting therefore the things which
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are close to it in some fashion in other words that are obtainable from it by just making small
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variants those are also likely to be interesting okay so um let's look at the Bernoulli numbers
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now we started with this exponential poly series x okay the usual power series or poly series for the
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exponential and this of course is important for many many reasons it plays an important role in
1:59
calculus its derivative is equal to itself so it's just a formal derivative you just take a final hop
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or a derivative term by term and you get the same thing back and of course then there's also the
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multiplicative relation that x plus a t times x plus bt equals x but a plus b t so these two
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properties sort of lie at the heart of many of the applications and the importance of the exponential
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poly series and i remind you once again that we're not thinking about this thing as a function
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okay that's not what it is not as in capital n o t okay it's not a function it is a poly series
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okay and then we recall that the Bernoulli numbers the b zeroes b1 b2 etcetera defined
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uh say this way here were obtained from this x by essentially moving this one to the other side
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and then dividing everything by t we get this new poly series 1 plus 1 half t plus 1 6t squared etc
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and this sort of variant can then be inverted if we invert just the or take the reciprocal of x t
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well then we just get x minus t because of this relation so the inverse of this thing is is very
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predictable but the inverse of this thing is much less predictable okay so when we take the inverse
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of that we get this uh series here um 1 minus a half t plus 1 12 t squared minus 1 over 70
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720 t to the fourth etcetera and there's an unpredictable aspect of this and when we uh
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sort of write it in the form summation b sub k over k factorial t to the k then that defines for
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us these Bernoulli numbers okay so that's a way of obtaining the Bernoulli numbers uh just sort of by
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making sort of manipulations at the power series level by starting with this obviously important
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thing x and then just making some smaller modifications to it and taking an inverse
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so i want to i want to carry on with that so um i'm going to do some more experimenting not in
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any particular direction just to see what happens okay all right so what i propose to do is to take
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the x series and subtract not just the constant term but the constant term and the linear term
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okay so let's go back up here the x starts with 1 plus t so if i subtract 1 plus t then i'm going to
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get a series that begins with t squared so there's a common factor of t squared in fact it's not one
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half t squared so if i then divide everything by one half t squared then i'm going to get a new
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series in t which starts with one one plus some linear term in t et cetera okay so that's what
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we're doing down here we're taking x of t minus one plus t and dividing by t squared over two to
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get this new series one plus one third t plus one twelfth t squared plus 1 60th t cubed plus 1 over
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360 t to the fourth and so on in some fashion and now in parallel to what we did uh with the
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Bernoulli story we're going to take the reciprocal of this we just take the reciprocal of this series
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and my computer will do that spell out the first few terms and it becomes 1 minus 1 3 t plus 1
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over 36 t squared plus 1 over 540 t cubed minus one over six thousand four hundred and eighty t
5:26
to the fourth etc and we expect this to be just as unpredictable if not more unpredictable than the
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Bernoulli numbers and in parallel to what we did with the Bernoulli situation let's just write this
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as a sum of terms c sub k over k factorial times t to the k for some new family of numbers
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call them c sub ks so these are just analog to the b sub case we've just sort of gone one
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step further in the the same direction that we did with the Bernoulli numbers okay so now it's
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not too hard to write down what some of these c sub k's are because we just compare this series
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here with with this one here c0 is equal to one then c1 is going to be minus one third and then
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c2 is going to be one over 36 times two or two factorial which is one over eighteen
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c three is going to be one over 540 times three factorials that's one over ninety
6:27
and uh carry on c4 will be minus one over and eighty 6480 four factorial which is minus one
6:35
over two hundred and seventy c five then is minus five over thousand one hundred and thirty four
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c six minus one over five thousand six hundred and seventy c seven equals 7 over
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2430 and we could clearly keep going so it appears as if these numbers are some kind of
6:57
variance of their Bernoulli numbers so how can we look up whether these are you know well known
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because they're rational numbers we have to look up their numerators or their denominators okay
7:12
that's the usual way we do things so in this case let's look at the denominators the sequence of
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denominators is 1 3 18 90 270 1 1 3 4 5 6 7 0. so we can look up that sequence in the online
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encyclopedia and lo and behold we get something we get sequence a double zero six five six eight
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which is denominators of generalized Bernoulli numbers so we have stumbled across
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in this fashion just by making sort of a random exploration from what we've already
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studied something that we're familiar with just making it sort of an obvious variant and we
7:58
stumbled upon these new generalized Bernoulli numbers which we didn't know about before
8:03
and carries on five six seventy and then two four three zero yes that's the right one okay
8:08
great all right so what does it say about these things well um so we have some papers arithmetical
8:16
properties of hypergeometric Bernoulli numbers by Daniel Iberhanu we paper on compositional
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Bernoulli numbers by Blandon and Diaz uh paper by Hassan and Nuen hypergeometric zeta functions
8:32
uh paper by Howard a sequence of numbers related to the exponential function
8:37
okay all of these look quite interesting and then a formula here Gary Adamson
8:45
contributes given a variant of pascal's triangle in which the two right-most diagonals are deleted
8:52
invert the triangle and extract the leftmost column okay this is very you know rings a bell
9:00
immediately because this is the same kind of thing that we've been looking at in terms of
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the pascal Harriet matrix and sort of stripping off the right most diagonal and then getting
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the Bernoulli numbers and in fact the foul harbor polynomials remember from last time so here we're
9:16
going to take up the suggestion so here's here's the same kind of array that we were using to
9:22
to invert to get the uh the Bernoulli numbers and the foul harbor polynomials okay and remember
9:29
that this one was obtained from the pascal Harriet matrix by basically stripping away the right most
9:34
diagonal of ones so let's get rid of the signs so just make everything positive and we'll strip
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off the current diagonal the one two three four five six etcetera so now the the diagonal just
9:47
consists of the triangular numbers 1 3 6 10 15 21 etc okay so this is yet another variant to the
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pascal Harriet matrix we've stripped off sort of two diagonals get this matrix c
10:03
okay and now we're going to take the inverse take c inverse and we get this highly unpredictable
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um or somewhat unpredictable you will shortly see there is something predictable about it
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highly unpredictable inverse matrix c inverse and down the first column just as that formula tells
10:25
us we recognize the same numbers that we just we've just observed these are these generalized
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Bernoulli numbers c sub k that we've identified so they're appearing right here
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okay so that's telling us that the the rows here are very likely to be interpretable
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somehow as generalized foul hopper polynomials because remember in the Bernoulli case these rows
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represented the foul harbor polynomials which
10:56
which solve this important problem of writing down formulas for sums of powers of natural numbers
11:05
okay so we should be able to stare at this and perhaps interpret rows here as also giving us
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generalized foul hopper polynomials which very likely you know have some application to
11:19
perhaps some kind of calculus question which calculus question i don't know
11:26
okay i don't know i think it's interesting in an investigation to make have a look at these
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polynomials see if you can find them or or interpret them or or see them as any kind of
11:38
variants of the foul hopper polynomials i remind you the following polynomials are hugely important
11:44
okay so if these are relatively close cousins of those then they're also likely to be interesting
11:54
and we make another sort of obvious observation here which is seemingly quite
12:01
remarkable um [Music] this second sub diagonal right one eight one over eighteen one over
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eighteen one over eighteen one over eighteen one over eighteen one over eighteen one over eighteen
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this is something that we haven't seen before
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this is you know why should this be why should it be
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that the second sub-diagonal is constant with the number 1 over 18.
12:30
and what about the checkerboard pattern of signs well if we look here we see some signs
12:36
but it's not they're not forming a checkerboard pattern it does appear as if every diagonal
12:43
is either positive or negative but the pattern of positives and negatives is um
12:52
not so clear i also remind you that when we were actually going back to the Bernoulli case we
12:56
actually didn't invert this matrix we actually put some minus signs we made a checkerboard pattern of
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science here before we inverted so you might want to investigate what happens if we do that as well
13:09
okay so it seems as if you know just by making this how should i say unmotivated exploration
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like an ai would okay here's something important so we're just going to systematically explore
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the area around it and the area around it is sort of defined by you know the complexity of
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the operation to get from the central object to anywhere else we want to do simple steps first
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okay the simple steps correspond to the to the nearer neighbors okay we have to do more multiple
13:43
complicated steps then we're going further away from this central object
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so that's sort of a way for us to to organize the landscape in terms of how complicated
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we have to make a variant from the original interesting object but we haven't done anything
13:59
too complicated to get this very interesting seemingly family of generalized Bernoulli
14:06
numbers and potential very uh interesting possible generalized final hopper polynomials now
14:16
just a quick google search on generalized Bernoulli numbers so i found this i didn't
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have a very systematic look but here's a book Bernoulli numbers and zeta functions by ibu
14:32
and they have a chapter here on what they call introducing generalized Bernoulli numbers
14:38
and Bernoulli polynomials now whether these generalized Bernoulli numbers are the same kinds
14:44
of things as are appearing here is a question okay it may be that they are doing something different
14:50
i suspect perhaps they are but this is something to to investigate you know what are these
14:54
generalized Bernoulli numbers um do they these these numbers that we're finding here these c's
15:00
these what are what are they good for what kind of properties do they have that are in parallel or
15:09
um you know in the same kind of direction as the many remarkable properties that are known for the
15:16
usual Bernoulli numbers so it looks like there's a huge potential for investigation here
15:25
and then so if we summarize so what did we do we we basically started with the exponential
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right and we did the same kind of thing as with the Bernoulli we but instead of just bringing
15:35
the one to the other side and then dividing by t we brought the one plus t to the other
15:39
side divided by by this next term okay well clearly those are just the first two steps
15:47
in a uh a sequence of similar operations that we can perform okay so once we've we've established
15:56
that you know we can we can take a step from the Bernoulli's to these generalized minerals well
16:00
then we can take a further step what would that involve it would mean we're going to take these
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three terms the one the t and the one half t squared to the other side
16:12
okay and then what's left is going to be divisible by 1 6 t cubed so we're going to divide everything
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by 1 6 of t cubed that's going to give us a new series that starts with one and then it has a a
16:26
t i guess one quarter t etc okay and then we're going to take that new series and we're going to
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invert it or take its reciprocal and we're going to get some new kinds of generalized
16:40
generalized generalized Bernoulli numbers we might call them d sub k's what do they look like
16:47
i don't know i haven't done that yet maybe you can do that and so maybe we see that you know
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there's in fact uh a whole family of such things that we could do right we could we could iterate
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this going up a level of complexity every time okay so we could you know after we've done this
17:04
one then we could go one step further and bring all these cubic terms to the other side divide by
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t to the fourth over 24 get a new series investigate its properties
17:15
have a new family of generalized generalized Bernoulli numbers say the e sub ks etc so
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just by making this kind of a simple-minded investigation okay we're potentially uncovering
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a rich structure that we then we want to you know put it down on the page and stare at it and then
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look at patterns and because i'm advocating a two-dimensional point of view here right
17:44
you can kind of see that we're um potentially looking at a Bernoulli array okay because we've
17:54
identified the Bernoulli numbers that's a sequence okay a sequence of numbers the bis b1 b2 b3 etc
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so now today i've talked about the c's i guess i guess c0 c1 c2 c3 etc okay
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and then i've suggested that there's going to be d0 d1 d2 so we can imagine sort of making an array
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that has all of these various sequences sort of in order giving us essentially a two-dimensional
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story okay so this is in keeping with our philosophy that we should
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try to think about arranging patterns as part of bigger patterns
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particular one-dimensional patterns trying to put them into two-dimensional patterns
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so here is a way of creating a two-dimensional pattern which contains the Bernoulli numbers
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prominently along the first row but potentially has all these you know new numbers with who knows
18:57
what uh interesting properties lots to investigate lots to think about and a lot of these things are
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accessible to amateur mathematicians so lots of opportunities for independent exploration.