Euler's Amicable Numbers - William Dunham

thank you yes this is about boiler and his amicable numbers and as I think I said it them at lunch if you were there I'm gonna break this into two parts a sort of introduction to Euler and then some euler's boy Larry in mathematics about amicable numbers so let us begin this is Euler you've probably seen this picture the official name of this is the Honda Minh portrait the unofficial name is euler emerging from the shower right it looks like it looks like he just took a shower if if someone was gonna come over and paint my picture I would not wear this I tell you but this is what he chose to wear so okay now let me give you a quick bio born in Basel Switzerland 1707 by the time he was a young teenager he was studying with Johann Bernoulli who happened to be in Basel at the time and this is quite fortuitous because Bernoulli was probably the greatest living mathematician at that point they're active mathematician Newton was still alive he was quite old liveness was gone Jakob Bernoulli his brother was gone so here is a great mentor for the young Euler he graduates from University of Basel in 1722 at the age of 15 his first appointment was to the st. Petersburg Academy in Russia in 1727 now you know that in Europe at this time the great courts had these academies collections of writers and musicians and scientists that were meant to enhance the reputation of the nation and the court and you know there was the Royal Society in London the Paris Academy the Berlin Academy and in Russia they wanted one they were just starting up the st. Petersburg Academy and they get oiler to come and be one of the early members so he was quite young at the time but it was a good choice he stays there until 1741 when he's lured away to the Berlin Academy under Frederick the Great and you know it was like he was a straight sports star that you know change teams or something so now now he's at Berlin where he meets Voltaire Dalembert but eventually he gets tired of that goes back to st. Petersburg and finishes his career in Russia and stays there until his death in 1783 now in terms of just a geographical life this is pretty constrained it wasn't it wasn't what he did in the physical world but in the mathematical world that made him so great I should just mention these dates are almost exactly those of Benjamin Franklin so if you want a comparison here on this side of the ocean boiler was almost exactly Franklin's contemporary although they never met they should have he's buried in st. Petersburg there's his tomb the story is he was originally buried in st. Petersburg then he was buried in Leningrad and now he's buried in st. Petersburg huh on the personal side he was married and had 13 children but the story is actually rather sad because this was the 18th century and childhood mortality was a very serious problem and of these 13 children only five survived to adolescence so it would have been a very sad household so much of the time oiler was blessed with a phenomenal memory all reports are that he could memorize anything books plays poems and tables so where are other people had to look up a logarithm he could just remember it and this would serve him well when physical infirmity strikes because in the 1730's he lost vision in his right eye and the the story is this was some kind of infection we think now that got away and now it could be cured with an antibiotic rather easily but it it not only took his vision it destroyed the eyeball so his eye just shrunk so it was of no use so he stops working right wrong he keeps going that doesn't stop him he keeps going at full speed until 1771 when he loses vision in the other eye and this was a cataract which is easily fixed today but not then and so what they tried was surgery and you don't even want to think about eye surgery in 17 yeah great Newton could bring his vodka yeah but you know they didn't have Newton but they had an eye surgeon but this was a horrible experience it was painful and it failed and so now Oilers essentially blind so he gives up right wrong his output does not diminish he keeps going just as strongly as ever and thereby becomes I think the great inspirational story in mathematics that he did not let the physical ailment stop him he would just come into a room and dictate to a table full of people writing furiously he would dictate his mathematics even though he couldn't see and you know he was using this memory where he had this ability to sort of see things in his mind's eye if a new paper came out from Dalembert or Lagrange they'd read it to him and he'd just absorb it and then make suggestions so you know he's like Beethoven you know Beethoven can't hear but he's writing great music boiler can't see but he's writing great mathematics and if you don't believe me that his productivity did not diminish I should point out that as a blind man in 1775 he published 50 papers a paper a week now that suggests what is a truism and that is that the quantity quantity and quality of Oilers work are almost hard to believe it sounds like science fiction and let me address these separately quantity of his work all right get this he died in 1783 but he wasn't done publishing in the years between 1783 and 1830 the what is it 47 years he managed to publish 159 papers but he still wasn't done he had a great year in 1844 when he published 61 and then there was a little blip in 18-49 when he published eight more so if you add these up he published 228 papers while dead no other dead mathematician has ever published that many papers you know it's amazing but these are things they found in his desk or things that were in press or or in his notebooks and they just kept coming to the surface and his productivity expanded in 1910 Gustave instrum decided to catalog all of Oilers published works and came out at 866 books and papers and Ennis trim you know dredges these up finds them and then writes little descriptions of each one little descriptions and the catalog is 388 pages long just the catalog describing Oilers work the swiss academy of sciences then begins publishing all the work and it's called the oprah omnia the first volume comes out in 1911 the volumes are like this they're big and substantial this is the one where the amicable numbers paper is but you can see that's that's heavy you know so this comes out in 1911 and then there's one in 1912 and they keep coming at the moment there are 75 volumes in four series and 25,000 pages and they're not done yet they're still coming out so this has 74 cousins and if you stack them it would go above the ceiling it's amazing the grandchildren of the original editors are old and they're still working on oil he's kept the swiss academy busy for a century and more there's there's nothing like it now in terms of quality well that's a little harder to measure perhaps and part of this will come out I think when I show you some of the things he does but here's one thing I did I went to math world you might know this this is a site and you can put in a term this is like a math dictionary and I put in oiler to see how many concepts in mathematics that are famous enough to get in the dictionary are attributed to him and I've came 96 entries from the oiler product some formula to the Euler identity to the Euler line to or Larian integrals and so on and so on that's a huge contribution of important ideas and just for comparison's sake Gauss I typed him in 70 Koshi a 33 and some people fare even less well math you have a question how many taught I do not know how many there are total no but but that's as far as I know that's the most things named after any one person okay so the quality is good now let me show you some of the things he does so here's one he gives us the number e in 1748 and here it is this is how it first appears it's he's writing he says for the sake of brevity we placed the letter E to be this number two point seven one eight two eight one eight two eight four five nine etc he loved calculating things out too many many places so there it is and he's going to call this e for the sake of brevity he says which therefore denotes the base of the natural or hyperbolic logarithm so he certainly knew what it was and if you want to calculate it he says here you go here's your formula you want to do this at home it's 1 plus 1 over 1 plus 1 over 1 times 2 plus 1 over 1 times 2 times 3 he didn't have the factorial notation but there it is that's e to the X where X is 1 and that converges very fast and allowed him to get this to a very high degree of accuracy now suppose he done nothing else this still would warrant a sentence or two you know in the math history books to give us e but he did so much more here's one the oil our identity I mentioned this at the lunchtime talk eetu the eye x equals cosine X plus I sine X and I showed you one of his proofs of that well here's how it appeared for the first time notice he didn't use I at this stage he was still writing square root of -1 and he was using v instead of x and he wanted to make this positive so he put a plus sign there I guess and curiously he put a period after cos and a sigh n because these were abbreviations cosine us sign O's and you didn't want to write the whole word I'd see you just abbreviated B had to put a period in we have now dropped the period but we have otherwise kept the and the notation in him and as everybody knows if you take that and stick in x equals PI you get I mentioned this today e to the I pi plus one equals zero always regarded as the often regarded is the most beautiful formula in all of mathematics because it has the five great constants of zero one e I PI all together in one formula the dream-team of constants all together you know right there it's amazing it's so amazing that somebody wrote a poem about it which I shall now inflict upon you so here it goes e to the I pi plus one equals zero made the mathematician Euler a hero from the real to complex with our brains and great flex he led us with zest but no fear oh ok so it ain't Shakespeare but it's ok ok what else well he gives us in 1752 the euler polyhedral form formula V Plus F equals Z plus 2 a polyhedron of course is a solid body a three-dimensional body whose faces are polygons like a cube and the vertices at faces he edges and if you look at the cube you see this works eight corners six faces think of a die 12 edges eight plus six equals 12 plus two okay that's pretty obvious but what a whaler says is this works for a vast number of solid bodies of polyhedra such as pyramids such as a casa Hedra and he proves this and this becomes you know one of the great results from Euler and a foundational concept in so much of later mathematics the oiler polyhedra formula so you know somebody wrote a book about this just the whole block about this formula a couple years ago published by Princeton so you know this is a biggie but oiler was a little more circumspect about it here was his review of this he said I find it surprising that these general results in solid geometry have not previously been noticed by anyone so far as I am aware how could anybody have missed this B plus F equals Z plus 2 he says well that's cause not everybody was alright he could see things that others just didn't okay there's the Basel problem another major achievement of his this is 1734 the issue is this in 1689 Jakob Bernoulli who was yo.hannes brother Johann was Euler's mad mentor challenge the mathematical community to find the sum of that infinite series the reciprocals of the squares what does that add up to and Jakob is writing from Basel Switzerland so this has came to be known as the Basel problem now I should say this that Jakob Bernoulli was no slouch with adding up infinite series he was very good and in 1689 he wrote something called the tract Otto's de serie abus infinities the treatise on infinite series in which he sums all kinds of sophisticated infinite series let me just show some K goes from 1 to infinity we didn't write it this way but this is our notation of K cubed over 2 to the K now when he's looking for exact answers not just yes this converges but exactly what it converges to and I should say that I've been all over the world and asked this question and nobody could answer this you know what this comes out to be and it's not 1 or 0 or infinity but the actual value of this as Bernoulli shows you're not gonna believe it 26 that's exactly what this ends up that's correct right so there's a problem for you if you want but but the point is that Bernoulli was good it's summing these series and you can sense his frustration that this series which looks considerably simpler he couldn't do he couldn't find the exact answer and he writes in the in the 1689 paper if anyone finds and communicates to us that which has thus far eluded our efforts great will be our gratitude you know please can somebody answer this for me well that's 1689 a generation passes nobody can until Euler comes along and in 1734 he gets the famous answer of PI squared over stick over 6 still one of the strangest formulas in all of mathematic now Euler does this by basically factoring the sine function writing in as an infinite product and an infinite sum and matching coefficients and that's his first proof but Euler being Euler he comes back and revisit sit and in 1741 he does it with integral calculus a very nice argument using the arc sine series and then he revisits its again in 1755 using differential calculus and three applications of low because roll to get this to fall out he finds a bizarre-o formula where if you put zero into one side you get the series and if you put zero into the other side you get zero over zero and he uses l'hopital's rule when he gets zero over zero and he uses it again and he gets zero over zero and most of us would have gone home by then but he uses it again and he gets PI squared over 6 it's an amazing argument and it just is a little aside here on pi day the math club here at Princeton has asked me to come and talk about PI the history of the famous constant March 14 and what I plan to do is talk about you know the crazy people involved in PI for the first half but for the second half I'm gonna derive this as Euler did with 3 uses of low betos role the most spectacular example of his symbol manipulation that I know ok so anyway that made him famous he was very young at the time he was in st. Petersburg nobody quite knew who he was before this but after he did this everyone in Europe knew this guy oiler the product some formula 17:37 here it is as it appeared in his journal in a journal I know if you can see this he has P equals 1 over 1 minus 1/2 times 1 minus 1/3 times 1 minus 1/5 times 1 minus a 7 he says that will be this P equals 1 plus 1/2 plus 1/3 plus 1/4 and if you write this in our notation what he's saying is the harmonic series is this infinite product over the primes and he gives a argument to justify this now wait a minute you say hold on this diverges this is infinity this is diverges this is infinity he's telling me infinity equals infinity here this is you know maybe a dumb thing to do a bad thing to do well not in euler's hands because starting with this he reasons in a brilliant fashion and addresses this question what happens if you sum the reciprocals of the primes converge or diverge I mean if you sum of the reciprocals of the squares converges the PI squared over 6 because there's not that many squares if you sum the reciprocals of the evens diverges what if you sum the reciprocals of the primes I don't think anyone had asked a question like this until then and it sounds very modern to me not not from 1737 well he takes it he applies that product sum formula and he gets the answer and you probably know it it's it's divergent the primes are sufficiently plentiful that that diverges and that's a neat fusion of number theory with primes and analysis and indeed Andre they said one may well regard these investigations as marking the birth of analytic number theory a subject which is now of course fabulously rich and and vast but it you know you could trace it right back to this ok now I want to get to my amicable numbers but just to other things real quick here's a curiosity this is not important but this is something or other did that I found interesting the challenge was to find for whole numbers the sum of any two of which is a perfect square now I have to amend this this is easy the numbers are 8 8 8 and 8 any two of them sum up to a perfect square so this should say different different whole numbers find four different whole numbers no matter which two you add you get a perfect square if you think that's easy try it you know it's not not so easy now here's what I think you know you know suppose I'm gonna do this just off the top of my head so I start with one okay now I need a number to be my second number add it to one to get a perfect square so I toss in three I need a number to add the three to get a perfect square so I toss in six but what happened you know now the 6 and the 1 don't add up to a perfect square so you go back and change this but then the ten and the three don't add up to a perfect square so you know you're like trying to juggle these things and it's hard too hard to do and you got to do four of them so if you try your you know your hand that there's just intuitively you're gonna be in trouble but Euler applies his great insight and he finds these four numbers one two three four and they work those numbers add up any two of them add up to a perfect square now you know you see this and you think this is magic this is crazy and then he tells you what he does oh yeah I see yeah I get it you know and it's not magic it's just oil ER you know but these work one last one before we get to my amicable numbers another curiosity everybody knows how to factor polynomials into first and second-degree factors right so x squared minus one breaks into two Linear's X to the fourth minus one breaks into a quadratic in two Linear's in the eighteenth century people were trying to prove this result any real polynomial of any degree can always be factored down into the product of real linear and or real quadratic factors and that's what the fundamental theorem of algebra looked like in the 18th century there was no complex numbers in it was just about real polynomials and of course you can't shatter them down into just linear pieces but linear and quadratic now so that was a challenge that was the conjecture you can do this for any polynomial and Euler tried to prove that but his proof failed actually so he wasn't perfect you know he got that wrong Gauss eventually proves it but somewhere in Euler's life time he gets a letter from Nicolas Bernoulli who said that's not true you can't factor all real polynomials in the first and second degree pieces because I have a fourth degree that you can't factor and Bernoulli sends us to Euler that one X to the fourth minus four X cubed plus 2x squared plus four X plus four says Bernoulli is irreducible can't break it down into two quadratics it's if that's right then the fundamental theorem is dead right well Bernoulli couldn't but that doesn't mean it can't be done oiler thought about he says I can factor this into two quadratics here's the first one and here's the second now you know you say there's two things two reactions one is this can't be right I mean these have square roots within square roots are you telling me if you multiply these two back together you get that simple quartic well I confess one afternoon I had nothing better to do and I multiplied these back together and it works this is this is right so this is right the second question is how on earth did he come up with this you know how do you do this there's this final proof that he's from outer space you know or something but then you read it and it's very clear he tells you what he does oh yeah I'd have thought of that you say yeah yeah I don't know it maybe he makes it seem easy but if there's a method there of course okay well so Euler was pretty good and he's pretty good in number theory when he comes upon this this concept of amicable pairs so the definition is that whole numbers m and n are amicable which means friendly right if each is the sum of the proper I should say whole number divisors of the other and by proper I mean smaller than the number so these are this is the concept friendly numbers amicable pairs now if you've ever seen this concept I'm sure this is the first example they've given this is the smallest one 220 and 284 now what I have to do then is take the proper divisors of 220 all the whole numbers that divided the 220 but aren't 220 and there they are 1 2 4 5 10 11 20 22 44 55 and a hundred and ten if you look on the yellow sheet you'll see them they're right there you have these up and you get to 84 now you take to 84 and you look at all of its proper divisors all the whole numbers is divided into 284 one to 471 142 now 284 does but that's not proper so that doesn't count you add those up and you get to 20 this guy mentioned in a minute yeah this goes way back so each of these is the sum of the proper divisors of the other it's a strange reciprocity it's kind of weird it's completely useless right there's no way this is gonna help you refrigerator run any better you know it's not gonna make money on Wall Street but it's what number theorists love right this kind of fascinating connection here true story I have a friend that's true he likes math when he got married he gave his wife a bracelet with the number 220 on it and he has a bracelet with the number 284 to symbolize their friendship you know forever they're they're lasting love and friendship it's really nerdy right but but it's really sweet right so some you know there's an idea for you if you if you need a present for somebody okay now to Peters question what's the history of these things well the Greeks knew this pair they knew 220 and 284 people think they just stumbled upon it it was probably just luck that they happen to hit upon this but you know remember the Pythagorean philosophy that whole numbers were sort of almost metaphysical the entities you know number is all we're gonna study these things almost worship these things so this was really perfect numbers also right that's right yes of perfect numbers if you're self friendly you're perfect right so so this was cool and the Greeks like this and they sought other pairs and they could find none and nobody could until the ninth century when the Islamic mathematician tobot even kora found a rule that generated two more pairs so now there were three the Greeks and Tommen's but apparently this rule did get back to Europe after the Renaissance so the people in Europe thought there was just the one pair as far as I can tell they didn't know about Tom it's work and so it was considered a great achievement when in 1636 Fermo finds a new pair seventeen thousand two ninety six and eighteen thousand four sixteen the sum of the property of visors of 17,000 to 96 is 18 for 1/6 and vice versa you can check that if you want you so Fermo finds this he's very proud of this he you know makes a big splash guess what that is one of table it's numbers he has just rediscovered something although he didn't know it but in fact he was going down the same path okay now in the 17th century France there was a great rivalry between Fermo and Descartes they did not like each other and here's Fermo coming up with a pair of amicable numbers and now it's like they cards honors at stake he's got to find a pair of amicable numbers so he goes to work on it and two years later he finds this pair nine million three hundred sixty-three thousand five hundred eighty-four and nine million four hundred thirty seven thousand and fifty-six each is the sum of the proper divisors of the other if you really love your wife you give her a bracelet with this number on it you know then Wow think about it well guess what that's tab it's other number Descartes was just following the same path that tobot him so these are the easy ones this is the low-hanging fruit of amicable numbers and after this for another century nobody can find any more and then Euler comes along in 1750 and in one paper he finds 58 so the world supply goes from 3 to 61 in one paper and the more you know he's multiplied the supply of these by 20 how'd he do it well that's what I'm gonna show ya but he sees a pattern he sees somethin that nobody before him had and it shows up in this paper and if you have your yellow sheet this is just copied this front page of this the title is the new Maurice Amica Billy bliss on amicable numbers and I put this on there mainly because I love saying the word Amica Billy buzz it's that is a great word right it brings a smile to your face if you ever feeling bad just say Amica Billy bus and you okay if you look here you see 220 and 284 showing up he's giving his little introduction and then he goes on to explain how he's gonna find dozens and dozens and dozens more so here we go on the math part the first thing we're going to do is introduce the oilers Sigma function this is where he introduces the Euler Sigma function in this paper and Sigma of n is going to be the sum of all the divisors of n including in he thinks that is a better way to go than just looking at the proper divisors now if you've had number theory you've seen this if you haven't well just let's do an example Sigma and 15 is the sum of all the whole number of divisors of 15 1 & 3 and 5 and 15 there they are you remember you count the 15 so that adds up to 24 this is a useful tool for characterizing primality P is prime if and only if Sigma P the sum of all the divisors of P is P plus 1 1 more than P so you get that it's so it has a head connection but it's also going to be a useful tool for characterizing amicability as well see Oyler realizes some things about it if for instance we have two different primes excuse me Sigma of the product is the product of the signals and again real quick if you haven't seen this proof it's quite simple Sigma of PQ you add up the divisors of PQ it there's not many 1 P Q P Q group the first two factor Q out of the second two factor 1 minus 1 plus P out of both and you get Sigma P Sigma of Q because these are primes and these are their signals so it's a it distributes across the product like that and so if I wanted to do that Sigma 15 the easy way I just say 15 is 3 times 5 and there's Sigma 3 which is 4 Sigma 5 is 6 and you get 24 but Euler knew more than that he knew that if the numbers were relatively prime even if they weren't Prime's themselves as long as they're relatively prime then the same property holds this is a multiplicative function Sigma of a b is sigma a sigma b and this is a tool for breaking down complicated numbers and finding their Sigma's so Sigma of 585 if I if I wanted that for some reason which I do you factor that in the 5 times 9 times 13 relatively prime factors split it in the Sigma 5 times Sigma 9 times Sigma of 13 Sigma 5 to 6 Sigma of 9 is 1 plus 3 plus 9 which is 13 Sigma 13s 14 and you get 1092 and I'm going to write that on the board here because we're gonna need that believe it or not in a minute so Sigma 585 is 1092 okay now we interrupt this lecture to bring an aside to you here I lied actually oiler didn't write Sigma of n the notation he used in his paper was that bad idea Euler you know I mean he was great with notation but you know people said no no that that symbol is sacrosanct you know we're not gonna use that for anything else so so they put in Sigma and that's good and here's another aside I just learned this the other day somebody told me this this is I hope this is right this is fascinating if you let H n be the partial sum of the harmonic series the nth partial sum of the harmonic series then this statement that Sigma of n is less than or equal to H sub n plus the exponential function of H sub n times the log of H sub n that statement for all n is logically equivalent to everybody know the Riemann hypothesis that is another version of the Riemann hypothesis and what's so striking about it is this is so simple relatively speaking you don't need complex numbers you don't need you know analytic extensions and you know the imaginary part the real part is 1/2 and on the the line all that stuff that's hard to explain to people this all lives in a very simple realm of Sigma Euler Sigma and the harmonic series partial sum and yet if you could prove this you'd get the Riemann hypothesis needless to say nobody's proved that but but anyway so you know Sigma is kind of still around yes that's right yes that's exactly what we're gonna be doing yeah right yeah so anyway that's so Sigma is still around but anyway back to Euler here's what he realizes now he's trying to characterize and me and we could build an amicability in terms of Sigma he says if these are the some of the proper divisors of the other the proper divisors of M are Sigma of M minus M because you know Sigma then includes the M and you don't want to and the proper divisors of n or Sigma of n minus n you don't want the N and so those two conditions are what amicability means but if you kind of put these together you get this very neat expression that Sigma of M is M plus N and Sigma of n is n plus m and so that is what he's gonna go after it if you can find m and n with that property there you've got your amicable numbers okay so he says here's what I'm gonna do he says I'm gonna assume a certain structure for these amicable numbers now later he could assume different structures but for this piece of the argument he's gonna say let's suppose the amicable numbers look like this M is a times P times Q and n the other one is a times R as I say we can we can expand this extend this later but for now this is the assumption suppose those are amicable and let's say that P Q and R are primes and different ones and they're all relatively prime to a so they have no common factor so you've got an a a times of P times of Q that's one number a times R is the other okay now remember that the condition Sigma of M equals m plus and equals Sigma of n so the first part of this he exploits is that just take the two n Sigma of M equals Sigma of n that would have to be true if these are going to be amicable stick in these values and you get Sigma of a PQ is Sigma of a R because that's what M and n are now use the multiplicative property the Sigma distributes or moves across here so you got Sigma of a Sigma P Sigma of Q is Sigma of a Sigma bar there go the Sigma of A's so Sigma P Sigma the Q is Sigma bar but P and Q and R Prime so Sigma P is P plus 1 Sigma of Q is Q plus 1 Sigma of RS r plus 1 so so the P Q and R have to satisfy that condition don't look elsewhere that's where you want to look there ok now he says let's try to clean this up a little bit so remember that P plus 1 times Q plus 1 is R plus 1 he's gonna let X be P plus 1 change change letters and Y be Q plus 1 ok so then it follows that P is X minus 1 and Q is y minus 1 okay and furthermore remember we we said that P plus 1 times Q plus 1 was R plus 1 so you put in here an x and a y and what you decide is that R is X Y minus 1 so now I so here's what's what these accomplish with this I'm gonna put a star there and it's important instead of P Q and R P now just his X and why he's lopped off a variable basically in his quest I still don't see any amigable numbers around ok now he says keep that in mind remember Sigma of M equals m plus and equals Sigma of n we exploited the exterior equality how about the the first one let's go back and remember that Sigma of M is M plus in see what we can squeeze out of that so M was a P Q and n was a R so we stick those in move the Sigma through on the left so you get Sigma of a times Sigma P times Sigma of Q and here it's not much I can do but factor the a out now I don't know I can't cancel the Sigma of a is this time inspect with this but Sigma P is P plus 1 because it P is a prime Sigma of Q is Q plus 1 so we're to that and let me put it up at the top of the screen here so there it is again so Sigma of a is times P plus one times Q plus one is a times P Q plus R but remember P was there X was P plus 1 Y was Q plus 1 n R was X Y minus 1 so it says get the peak use or out of there and get the X's and Y's in so here's where we are now Sigma of AP plus 1 is the XQ plus 1 is the Y here's the P X minus 1 here's the Q and there's the R X Y minus and now inside that square bracket simplify and you'll see you get an X Y minus an X minus a y plus a 1 and then plus XY minus 1 so that condenses to 2xy minus X minus y and the ones cancel ok now what well now he wants to solve this for y so if you look at this there's almost all the terms have a Y in it this minus ax doesn't so that goes to the right to the left and then what's what's left is a to a XY minus a Sigma ax y minus and a Y and he factors the Y and then solve for y by dividing by that coefficient and we get this Y is ax over this guy but in this one he he factors the X out of the first two so he gets Sigma of a or to a minus Sigma of a times X and then minus a and I still don't see any amicable numbers okay but we're getting there we're getting there so let me move that up to the top now time out here for a breather let's see at this point he has introduced m n P Q R X & Y but hey there's still letters left you know the alphabet has lots more lighter so it's time for more letters so he does this all the time his mathematics he's always substituting a new variable for old so he says here's what I want to do look at a over to a minus Sigma they now now that's going to be a whole number over a whole number reduce it to lowest terms and call that B over C new lighters B&C I think these are the last of the letters you need okay now if that's the case then 2a minus Sigma of a is a C over B just solve for that and put that in the square bracket there so why is ax over instead of the thing in the square bracket I'm going to have a C over B times X minus a there go the A's every term is an A in it white mom and you'll be left with x over what's C over B X minus 1 when you divide out the a multiply that through by B to clear fractions and you get that Y is BX over C X minus B so there's what I in terms of the X and the B and the C and believe it or not there's one more thing he needs and then he's ready and then one more thing he needs this I would have never thought of this of course I would have never gotten anywhere here but he says let's look at this see y minus B that's the thing he really wants see Y minus B now that's going to be C times y but we just did Y up here so you put in VX / CX - because that's Y and then minus B common denominator here is CX minus B if you look at this you've got a BC X minus BC X plus a B squared and so CA C Y minus B turns out to be B squared over C X minus B and what he does with this is cross multiplies and finally we got the big equation C X minus B times C y minus B is B squared and I'll put a double star there that's right it's the numerator yeah right yeah that's right that's right so it's complicated but you get down to this and he said this is the key because when I do this I'm going to do one in just a minute B and C are gonna be actual numbers and B squared is just a number and if it's the product of two things there's only a few ways that can happen you know a limited number of ways you just check them until you find x and y that build back to give you P Q and R that are primes and you get your amicable numbers so you know he sees this as the salvation of this problem okay now so let me do one so now the issue then is where do you start so let's let oh I guess I lied here let me give you the strategy okay then I'll get to my example now this is on the back of the yellow sheet if you want to follow along cuz it's pretty complicated you start with a you then calculate a over to a minus Sigma they reduce it to lowest lowest terms you get B over C you use the double star equation CX minus B times C y minus B is B squared so you can find x and y then you can find P Q and R and if they're all prime you put them back together again and build your amicable numbers M as a PQ and n is a R that's his strategy so it all starts with a so now okay here we go I'm gonna start with a equals four so the first thing you got to do is calculate a over to a minus Sigma V which is 4 over 8 minus Sigma for the sum of all the divisors of four seven one plus two plus four so you get four over one and that already is in lowest terms so B is 4 and C is one okay now you go according to the y'all sheet you go to the this guy CX minus B times C y minus B is B squared I put in C equals 1 and B equals 4 and you get X minus 4 times y minus 4 is 4 squared 16 okay now he actually makes a little chart now as he starts hunting for these guys so I will do the same so there's the critical expression X minus 4 times y minus 4 is 16 and here's what you got you gotta fill in the chart to find x and y and PQR and you're looking remember for Prime's P Q and R in practice alright so the first thing is what two numbers multiply to be 16 that's what the X minus 4 and the y minus 4 well let's start with 16 + 1 16 times 1 is 16 and now you start filling in now if X minus 4 is 16 X is 20 if y minus 4 is 1 Y is 5 all right now now you go after the primes so P is 1 less than X so it's 19 prime Q is 1 less than Y it's 4 and not a prime right so this is out this doesn't work now most of them don't work this is a dry hole most of the things he tries you just throw away you know you're looking for those golden nuggets where it all works this time it did okay so you try different a different product what else multiplies to be 16 well how about 8 times 2 so 8 times 2 alright now we played the same game again if X minus 4 is 8 X is 12 if Y minus 4 is 2 y is 6 P is 1 less than X that's 11 prime Q is 1 less than 6 that's 5 prime this is good but we're not done yet we got to do our this has to be prime so that's X Y minus 172 minus 1 is 71 yeah prime so now you've got it this is what you're after and now what you do is build your numbers back M is a PQ and as a our a was the seed for and so here we go M is for a times P 11 times 5 and n is for the seed times 71 and you multiply these out you get 220 and 284 good news bad news right good news they're amicable bad news is that's the one they already knew so that's not a new pair but at least it shows that we're on the right track now if you start with the seed of 16 you get pheromones pair if you start with the seed of 128 you get the card I don't think so that's a very good question yeah I probably not know that would be interesting but notice what Fairmont and Descartes and indeed what tobot were doing they had to hit upon a rule that works with powers of 2 2 squared 4 4 2 to the fourth 2 to the 7 but then there's no others that work that are within in any any reasonable size than anybody could have done back then for the powers of 2 and so they were done but Euler never assumed they was a power of 2 and his whole theory is much more general and that's where he unlocks the secret so here comes a new one so now now pretend like it's you know 1750 and all you know are those three pairs I'm going to show you how he finds a brand-new one and he uses 585 this is seed ok so what do we do first we have to figure out a over to a minus Sigma of a so that's 585 double that 1170 minus Sigma 585 I've seen that somewhere oh there it is 1092 right we did that so stick that in there and you get 585 over 78 which isn't in lowest terms because 39 goes into both of those so you reduce it to 15 halves so B is 15 and C's - okay now you look at this see X minus B times C y minus B is B squared where C is 2 and B is 15 and so that becomes 2x minus 15 times 2 y minus 15 is 15 squared - 25 so this is the critical equation I put it at the top I make my little chart and here we go hunting for primes all right first thing you can try is 225 times 1 now follow me on this 2y minus 15 is 225 so 2y is 240 so Y is 120 I said I mean excuse me X is 122 y minus 15 is 1 so 2y is 16 so Y is 8 back up one 120 minus 1 is 119 prime no right it sure looks like a prime right it's got a nine on the end but it isn't it's the visible by seven times seventeen yes so did dry it dry hole there that didn't work okay back to the drawing boards how about 75 times three if 2x minus 15 is 75 2x is 90 X is 45 I think I see trouble coming already but let me do the other one here 2y minus 15 is 3 2 y is 18 y is 9 this is big trouble as soon as I go after 45 minus 1 dry hole okay how about 45 times 5 that's another way to get to 25 2x minus 15 is 45 2x is 60 X is 30 mm promising 2 y minus 15 is 5 2 y is 20 wise 10 mmm 30 minus 1 is good but 10 minus 1 is it so that falls through now you say hey wait I'm sitting here it's late show me an example where it works oh this is gonna work the next one I promise you will work so the next one is 25 times 9 if 2x minus 15 is 25 2 X is 40 X is 22 y minus 15 is 9 2 y is 24 wise 12 back off from the 20 you get 19 prime back up from the 12 you get 11 prime but there's still one more X Y minus 1 which is 240 minus 1 which is 239 and thank goodness that's a prime too so we're in business and now Euler builds his amicable numbers remember the seed was 585 the a so M is a PQ 585 times 19 times 11 120 mm 265 and n is a are 585 times 2 3930 9815 that's an amicable pair and when he wrote these down no one on earth had ever seen them that was brand new and he did it as I showed you which is elementary I mean nothing here requires anything very sophisticated except an IQ of 4 digits or whatever he had so this is how he does it and then he does more and more and more of these and if you want to do one of these at home here's some seeds that he likes you got 90 - and 819 and 57:33 these a little bit yeah there's certain things he doesn't bother trying but these work if you want to try them and it would you know they'll be I'd pay off in other words but then that didn't yield 58 new ones he then goes back and re jiggers the M&N he says well what if M is a PQ and then is a RS where these are primes now what conditions follow and what new pairs going to get and one of em is a PQ R and N is a st and he gets some more and by modifying the the structure of these he builds up his collection to blow the problem out of the water as I say well I don't know I read this I'm impressed I think this is pretty good you know for somebody so long ago to be doing something so insightful so there's just two slides left the one is the woman might my first reaction when I saw this was I make a billing list right you know I got to say my word and yeah okay so good good for him all right and the second thing is what I find myself saying often whenever I read one of these things by Euler and I'm left dumbfounded way to go uncle I&R good work thank you so yeah so Peters question is what's the status of nobody knows if there's infinitely many but what they have done of course is turn yes they've turned the computer on and I met a guy once who whose PhD he said was finding 10,000 more pairs with the computer you know and I said how did you do it and he said basically that way but you see oiler got stuck what what eventually limited him was if you want to take Sigma of a you got to break a down into primes and they didn't have they didn't know which numbers were prime I think I'm not sure but I think they think there's infinitely many but you know there's certainly tens of thousands that have been identified I'm gonna say so oh yes I gave this talk a similar talk once at Baylor and at the end there was this little old guy sitting in the bag and he came up to me and he said you know when I was very young I worked on this and and he was at Vanderbilt and he commandeered their computer overnight now their computer in those days you know it was like tubes you know and it filled two rooms but he had it looking just by brute force he didn't do the theory he just started looking at the sums of the proper divisors and looking for matches and he found the largest five digit pair so they're both five digits and with his computer search in 1958 or something overnight he found the largest pair so I thought wow this is history you know right in front of me you know there's we can go back to oil or we can go back to the Greeks but there's this guy at Baylor who who was still working on it so there's probably infinitely many but nobody's proved that nor is anybody found a use for it yeah but that's okay yeah my numbers like 1 no and so the question is this was quite amazing was the 19th century right and this little kid finds two four digit ones and you're right it was 11:59 and something that no one had spotted including oil it fell through his sieve here he didn't catch it and so this was astonishing you know this this kid found these four digital ones that everyone in missed and so then the question is what happened to the kid you know did he did he win the first Fields Medal and the answer is no he even became a shoemaker or something that was it that was his one mathematical for a but he made a big splash by finding a to four-digit numbers now they know all the ones you know there's not gonna be any surprises below a certain number they checked but but that one fell through Euler's yet Matthew yes yeah Fredrick the great Frederick yeah he whenever oiler went to the Berlin Academy Frederick the Great hired him and so yes there was a I mean Frederic knew him they talked to each other Frederick didn't like him amazingly enough Frederick Frederick was a Francophile right a sophisticate he like Voltaire he liked philosophy Euler was a meat and potatoes kind of guy from Switzerland you know he didn't like that stuff and so they didn't get along they didn't hit it off boiler was extremely devout he was a devout Christian and Frederick wasn't and Voltaire certainly wasn't and so you know there was friction there and the the senses Frederic regarded Euler is a country bumpkin you know unsophisticated although he could fill more pages with mathematics than anybody else but he wasn't in in Frederick's league he never met Bach however so it was a good time that I don't know the question was did Voltaire dislike Euler and I'm not sure they were both at the Academy how much contact they had I don't know but I do know Frederick probably you know when you're at the Academy and the monarch says do something you have to so so sometimes Euler would be taken away from his math ago you know design a fountain or something for the palace and you know that's what that's what they wanted him to do Frederick was actually quite cruel to Euler you know Euler's vision has gone and Fred we took the calling him my Cyclops which is really quite awful right so there was a real friction there and not a respect and that's one of the reasons oiler went back to Russia he was much more welcomed there and got paid a gigantic amount of money when he went back so yep I don't know that Katherine did yeah have you heard that that she did yeah I had heard no I had heard that coming from Frederick but that perhaps wasn't nice whatever thank you

Channel: princetonmathematics

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Keywords: Princeton University, Mathematics, Leonhard Euler (Author)

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Length: 56min 55sec (3415 seconds)

Published: Mon Mar 03 2014