Amicable umbers November 005 How Euler Did It by Ed Sadifer Six is a special umber. It is divisible by, ad 3, ad, i what at first looks like a strage coicidece, 6 = + + 3. The umber 8 shares this remarkable property; its divisors,,, 4, 7 ad 4, sum to the umber 8. Numbers with this property, that they are the sum of their divisors (icludig, but ot icludig the umber itself) have bee kow sice aciet times ad are called perfect umbers. Euclid himself proved i Book IX, propositio 36 of the Elemets [E]: If as may umbers as we please begiig from a uit be set out cotiuously i double proportio util the sum of all becomes prime, ad if the sum multiplied ito the last make some umber, the the product will be perfect. I a more moder treatmet, Hardy ad Wright [H+W] state this same theorem as THEOREM 76: If + is prime, the ( ) + is perfect. Each such perfect umber is associated with a prime of the form +, ad such umbers are ow called Mersee primes. Several Mersee primes are kow, ad for several decades, the largest kow prime umber was usually a Mersee prime. This is o loger the case. Euler proved that all eve perfect umbers have the form i Theorem 76, ad also discovered a few properties that a perfect umber would have to have if it were odd. Sice o odd perfect umbers are kow, it is difficult to explai to o-mathematicias why it might be iterestig to prove thigs about them ayway. As far as I kow, the two best-kow properties of odd perfect umbers are:. There might ot be ay, ad. if there are ay, they must be very large. But we are off the track of the story. Cosider the pair of umbers, 0 ad 84. The divisors of 0 are,, 4, 5, 0,, 0,, 44, 55 ad 0, ad those divisors sum to 84. Meawhile, the divisors if 84 are,, 4, 7 ad 4, ad they sum to 0. Such pairs of umbers, the divisors of oe summig to the other, are called amicable pairs. For over a thousad years, oly this pair, 0 ad 84, was kow. Iamblichus, i the fourth cetury BCE, wrote, The first two friedly umbers are these: sigma pi delta ad sigma kappa. I the
Greek umber system i use at the time, sigma had a value 00, pi ad kappa were 80 ad 0 respectively, ad delta was 4, so he was describig 84 ad 0. I the 9 th cetury, Arab mathematicia Thabit ib Qurra probably discovered the ext amicable pair, 796, 846. I the 600 s, Pierre Fermat rediscovered this pair, ad his mathematical rival Reé Descartes discovered aother pair, 9,363,584 ad 9,437,056. So, whe Euler came o the scee, oly three pairs of amicable umbers were kow. The, i 747, Euler published a short paper [E00] metioig the techique that Descartes ad Fermat had used, ad listig 30 amicable pairs, icludig the three already kow, ad icludig oe pair that was ot actually amicable. Nevertheless, i oe paper, Euler legtheed the list of kow amicable pairs by a factor of almost te. Euler gives us almost o clue about how he foud these umbers. He briefly describes the methods Descartes ad Fermat had used, though. They had cosidered pairs of umbers of the form xy ad z, where x, y ad z are all prime, ad showed that, for the umbers to be a amicable pair, it was ecessary that z = xy + x + y. Fermat ad Descartes had just searched for prime umbers x, y ad z to see which oes gave amicable pairs. However, this caot be how Euler foud his ew amicable pairs, sice oly the first three, the oes that were already kow, have this form. Eleve of the others have the form xy ad zw, where x, y, z ad w are all prime, but others ivolve as may as seve distict prime factors, ad te of the pairs are pairs of odd umbers. It is ot like Euler to leave us i the dark like this, without showig us how he made his discoveries, ad I ca offer o very satisfyig explaatio. It is true that most articles published i the Nova acta eruditorum were rather brief, but this article was oly three pages log. A author of Euler s stature would have bee welcome to write six or seve pages, if he had wated to. It is also true that few importat mathematicias had worked o umber theory sice the days of Fermat, who died i 665, 80 years before Euler wrote this article, ad this was oly Euler s sixth article that the Editors of the Opera Omia classify as umber theory. Sice Euler published over 90 such articles, E00 comes quite early i his umber theory career. Neither of these seems to explai why Euler chose to be so obscure. Later i 747, though, Euler wrote aother paper, Theoremata circa divisores umerorum, or Theorems about divisors of umbers, [E34] i which he explaied how he had discovered that the 5 fifth Fermat umber, + was ot prime, but was divisible by 64, ad also gave his first proof of Fermat s Little Theorem. This was the subject of the very first colum i this series, back i November of 003. Perhaps that paper got Euler thikig about providig better explaatios of his discoveries i umber theory, or maybe it just kept him iterested i umber theory. Whatever the reaso, i 750, Euler retured to the problem of amicable umbers, armed with a powerful ew idea, the first of what we ow call umber theoretical fuctios. He iveted a ew fuctio ad a ew otatio, deotig the sum of the divisors of a umber, icludig itself, by The itegral sig is supposed to remid us that we are summig somethig. This fuctio is ow sometimes called the sigma fuctio ad deoted σ ( ). Here we will use Euler s otatio. Immediately after itroducig his ew otatio, Euler gives the example that. 6= + + 3+ 6=, ad that, i geeral, perfect umbers are those for which = ad prime umbers are those for which = +. He pays due attetio to his fudametal case, =, ad otes that this shows that the uit ought ot be listed amog the prime umbers. He follows with a expositio almost idistiguishable from that i a moder umber theory textbook, of the basic properties of his ew fuctio:
Lemma : If m ad are relatively prime, the m= m Corollary: If m, ad p are prime umbers, the mp= m p= ( + m)( + )( + p) k k k + Lemma : If is a prime umber, the = + + +... + =. a β γ δ Lemma 3: If a umber N has a prime factorizatio N = m p q etc., the α β γ δ N = m p q etc. Euler does a few examples like fidig that 360= 70 ad usig his ew fuctio to show that 60 ad 94 form a amicable pair. With this last example, he is showig off a bit, sice this pair is ot amog the three amicable pairs kow i aciet times, though it was o his list i E00. The he turs to characterizig amicable umbers. Here is how he does it. If m ad are amicable pairs, the m m= ad = m, ad a little bit of algebra leads to the form Euler wats: m= = m+. Armed with this, he begis to study amicable pairs that share a commo factor, a. He classifies these as follows: third form first form apqr as apq ar fourth form secod form apqr ast apq ars fifth form apqr astu A moder reader might wat to cout the factors that the pairs do ot have i commo, ad the classify these with a otatio like (, ), (, ), (3, ), (3, ), (3, 3), etc. We could the try to make a case that it resembles Cator s diagoal proof that the ratioal umbers are coutable, but such observatios are aachroistic, ad are more amusig tha they are useful or valid. Now he cosiders these oe form at a time. PROBLEM First, Euler cosiders amicable pairs of the form apq ad ar, where there is a commo factor a ad the umbers p, q ad r are prime umbers ad ot factors of a. All of the amicable pairs kow before Euler s time were of this form ad had a beig a power of. The coditio he foud earlier implies that r = p q, ad, sice p, q ad r are prime, this meas that r + = (p + )(q + ). Substitutig x for p + ad y for q +, this makes r = xy, where the umbers x, y ad xy must all be prime, ad the umbers a(x )(y ) ad a(xy ) form the amicable pair he is seekig. Moreover, the coditio m= = m+ becomes 3
( ) a xy x y = xy a or y = ax. a a x a ( ) b a He simplifies this with the substitutio =, b take to be i lowest terms. c a a c Substitutig this ito the expressio for y, he gets the fairly simple form ( cx b)( cy b) = bb ad, because p, q ad r are prime, he gives the additioal coditios that x, y ad xy must all be prime. This is eough ew iformatio to start searchig for amicable pairs. He begis what he calls Rule, ad supposes that a is a power of, say a = k. His substitutios lead to b = ad c =, so that x y =. ( )( ) Euler did t leave out ay steps of this calculatio, but i Euler s day, paper was expesive, ad we have a choice of cheap paper or computer algebra systems if we wat to check his work. Cotiuig, there are t very may ways to factor, ad this product must have the form for some value of k. From this it follows that ( x )( x ) + = + k k + k x = + k y = + ad the three prime umbers p, q ad r that to i to makig the amicable pair are + k p = x = + k q = y = + + + k k r = xy = + + Euler makes oe more, i this case rather uecessary substitutio, takig m = k, so that = m + k, ad rewrites these equatios i terms of m ad k istead of ad k. We ll skip that. Now, Euler cosiders as separate cases various values of k. First, if k =, we look for primes of the forms m p = 3 m q = 6 m r = 8 If m =, the these give prime umbers 5, ad 7, ad so the umbers 0 = 5 84 = 7 is a amicable pair. If m =, we get the umbers, 3 ad 87. The first two are prime, but the third is 7 4, ad so this case does ot yield a amicable pair. If m = 3, we get the umbers primes 3, 47 ad 5, ad hece the amicable pair 4
4 7,96= 3 47 4 8,46= 5 After a few more ufruitful substitutios for m = 4 ad m = 5, takig m = 6 gives aother amicable pair, which we will leave to the reader to calculate. Euler also cosiders cases k =, 3, 4 ad 5, but they yield o additioal amicable pairs. He assures us that these three are the oly amicable pairs of this first form ivolvig a commo factor of ad ivolvig oly prime umbers less tha 00,000 He further cosiders, without ay positive results, commo factors of the form + k a = +, for which the secod factor is also a prime umber. Euler calls this secod prime ( ) factor f, ad with a calculatio almost exactly like the oe above, cocludes that for a amicable pair to + m be geerated, there must be expoets m ad, for which x = + ad + m+ + m y = + + for which m < + ad all four of the followig umbers must be prime: ( )( ) f = + + m+ p = x q = y r = xy This is as far as Euler ca go here with aalysis, so it is time to examie cases. Takig m = yields o amicable pairs. However, if m =, it makes + f = 3, x 3 + y = 3 3 ad a = f = ad ( ) whece + + p = 3, q = 3 ( 3 ) ad r = 9 ( 3 ). Oe eed oly substitute various values of, hopig to make all four of the umbers f, p, q ad r prime. Euler does this i a table: = 3 4 5 f = 3 47 95* 9 p = 5 47 q = 3* 37 563 967* r = 98* 87 6767* I this table, umbers that are ot prime are marked with a *, ad the ellipses mark umbers that were uecessary to calculate because there is already a composite umber i that colum. The 98 i colum was uecessary, but easy, so Euler did it ayway. Oly colum is free of composite umbers, ad this meas that it leads to a ew amicable pair: 43 537 43 87 This is the first ew amicable pair that Euler has show us how to fid, ad it was the first ew oe o his list back i E00. After this, the fu is over, eve though the paper is less tha half fiished. Euler cotiues for aother 50 pages, doig more forms, more cases, ad turig up more ad more amicable pairs. At the ed of the paper, he summarizes his results, givig 6 amicable pairs, with a couple of typographical errors ad a couple of mistakes, ad doublig agai the world s populatio of kow amicable umbers. Rather tha slog through this, we ll leave it to the iterested reader. 5
Euler wrote a third article with this same title, De umeris amicabilibus [E798]. He did t fiish it ad it was ot published durig his lifetime, but was foud amog his papers ad published i 849, more tha 60 years after his death. It is more pedagogical tha the other two papers, ad he does ot give ay ew amicable pairs. Amicable umbers are a curious topic. Euler s methods succeeded i reducig a early impossible search for special pairs of umbers to a more maageable search. He could t guaratee that umbers of a particular form would be amicable, but he made the search small eough that he was able to fid quite a few of them. Now, usig Euler s methods, but usig computers to do the gigatic calculatios, thousads of amicable pairs are kow. Refereces: [E] Euclid, The Thirtee Books of Euclid s Elemets, ed, Sir Thomas Heath, tr., Cambridge Uiversity Press, 96. Also available as a Dover reprit. [H+W] Hardy, G. H. ad E. M. Wright, A Itroductio to the Theory of Numbers, 3ed, Oxford Uiversity Press, 954. [E00] Euler, Leohard, De umeris amicabilibus, Nova acta eruditorum, 747, p. 67-69, reprited i the Opera Omia, Series I volume, p. 59-6. Also available o lie at www.eulerarchive.org. [E34] Euler, Leohard, Theoremata circa divisores umerorum, Novi commetarii academiae scietiarum Petropolitaae, (747/48), 750, p. 0-48, reprited i the Opera Omia, Series I volume, p. 6-85. Also available o lie at www.eulerarchive.org. [E5] Euler, Leohard, De umeris amicabilibus, Opuscula varii argumeti, 750, p. 3-07, reprited i the Opera Omia, Series I volume, p. 86-6. Also available o lie at www.eulerarchive.org. [E798] Euler, Leohard, De umeris amicabilibus, Commetatioes arithmeticae, 849, p. 67-636, reprited i the Opera Omia, Series I volume 4, p. 353-365. Also available o lie at www.eulerarchive.org. Ed Sadifer (SadiferE@wcsu.edu) is Professor of Mathematics at Wester Coecticut State Uiversity i Dabury, CT. He is a avid maratho ruer, with 33 Bosto Marathos o his shoes, ad he is Secretary of The Euler Society (www.eulersociety.org) How Euler Did It is updated each moth. Copyright 005 Ed Sadifer 6