How Euler Did It. by Ed Sandifer. Theorema Arithmeticum. March, 2005

Transcription

1 How Euler Did It Theore Aritheticu Mrch, 005 by Ed Sdifer Euler s 748 textbook, the Itroductio i lysi ifiitoru, ws oe of the ost ifluetil thetics books of ll tie Joh Blto s excellet trsltio, is vilble i y librries, d o the bookshelves of those few idividuls who were lucky eough to get copies i the reltively brief tie it ws i prit Those people see to be holdig o to their copies, s there see to be lost oe vilble o the used book rket If you red the Itroductio, you re likely to hve differet rectios to differet prts of the book Whe you red the sectio bout prtil frctios or the sectio bout the defiitio the trigooetric fuctios, you will feel very uch t hoe Euler s tretet is very siilr to the wy we preset these topics tody This is becuse, for this topic, how Euler did it bece dopted s the stdrd wy to do it Whe you red bout series, you y feel like Euler is doig soe thigs the hrd wy becuse he does t use clculus This is preclculus book, itroductio to the ethods d teril tht will be used i clculus, so he does ot use Tylor series or other clculus tools It is surprisig how uch he is ble to do without clculus Soe other topics, like prtitios d cotiued frctios, re t see so ofte y ore, d it is excitig to see how uch c be doe by eleetry es The Itroductio provides kid of foudtio for uch of Euler s creer Tie d tie gi he fids le i the Itroductio tht he eeds i soe lter pper, or he writes whole pper tht begis with topic fro the Itroductio This oth s colu, though, is t bout the Itroductio It is bout soethig tht would hve fit well with soe of the other teril i the Itroductio Mybe it should hve bee there Whe Euler eeded this little result, it ws t there, so he hd to puse to prove it Euler wrote ssive text o clculus His Istitutioes clculi differetilis, E-, ws published i 755, seve yers fter the Itroductio More th te yers lter, i 769, E-34, E-366 d E-385, his three volues of the Istitutioes clculi itegrlis ce out At ore th 500 pges, these four volues outweigh eve the ost prolix of oder texts Though Euler ws seldo ccused of beig too brief, we should deflect soe criticis of his verbige; he does iclude both extesive tretet of differetil equtios d good bit of the clculus of vritios uder his ubrell of clculus The whole clculus series is preseted s series of probles Book I of the Clculis itegrlis, for exple, hs 73 probles, spred cross two volues Ech proble is give i rther geerl

2 for, d with geerl solutio Most solutios re followed by uber of corollries, scholios or exples Ech proble, corollry or other prt hs prgrph uber, though ost cosist of ore th oe prgrph Book I hs 75 such prgrphs Our exple coes fro er the ed of Book I of the Clculis itegrlis, prt of volue of book I, so this is foud i E-366 It follows Proble 5 d is i prgrph 69 At this poit, Euler hs bee doig differetil equtios for over 300 pges He coes upo rther coplicted proble (there is t spce to get i to it here) tht c be drticlly siplified usig clever prtil frctio expsio Norlly t this poit, Euler would refer to the Itroductio to fid the le tht solves the proble This tie, the result is t there! So Euler puses to give us: THEOREMA ARITHMETICUM Give ubers, b, c, d, etc, if fro ech oe is subtrcted ech other oe d the followig products re fored: the it will lwys be tht ( b)( c)( d)( e) ( b )( b c)( b d)( b e) ( c )( c b)( c d)( c e) ( d )( d b)( d c)( d e) etc = α etc = β etc = γ etc = δ etc = 0 α β γ δ Euler overlooks the coditio tht the ubers, b, c, d, etc, ought to be distict, or else two of the products will be ero d the forul i the coclusio will be udefied If we hve three ubers,, b d c, the Euler is cliig tht + + = 0 ( b)( c) ( b )( b c) ( c )( c b) With bit of lgebr, the reder who is creful with sigs c esily verify this idetity by usig ( b)( c)( b c) s coo deoitor The cse of four ubers, though, would require coo deoitor with six fctors, b c d b c b d c d, d the lgebr is cosiderbly ore cubersoe I ( )( )( )( )( )( ) geerl, ubers would require = fctors This quickly oves fro the wkwrd to the ifesible There ust be better wy The oder reder would probbly rewrite Euler s cli usig subscripts, sigs d product sybols Let the ubers be,, 3,,, d the products be give by αi = ( i j) The Euler clis tht = 0 The, with creful geet of subscripts d sybols, it is i= αi probbly possible to prove the result It would probbly ot see clever Euler though, did ot hve those tools, so he hd to fid clever wy j= j i

3 Euler begis his proof with step tht kes the reder expect proof by theticl iductio Tht s ot wht he s doig, though He supposes tht the lst of his ubers is deoted by, d tht Z is polyoil i of degree less th He fors the rtiol expressio Z ( )( b)( c)( d) etc Note tht the deoitor here will be the lst of the products Euler defied i the stteet of his theore, so tht ζ = ( )( b)( c)( d)etc Now, sice he kows his Itroductio, he decoposes this ito its prtil frctios: (*) A + B + C + D + etc b c d We will eed the egtive of this expressio, so Euler otes tht its egtive is (**) A + B + C + D + etc b c d He is relly oly iterested i the specil cse where Z =, where is less th For this prticulr Z, we c do just little work, we c fid the ubers A, B, C, etc explicitly s A = B = C = ( b)( c)( d) ( b )( b c)( b d) ( c )( c b)( c d) etc The lst fctors of these deoitors re ( ), (b ), (c ), etc Sice Euler will be iterested i the expsio of, he will be usig frctios for which the deoitors ivolve ( ), ζ ( b), ( c), etc, hece his rerk tht gve us the equtio rked (**) With this groudwork set out, Euler is redy to look t the proble itself Tkig y to deote the peultite ter, the products give i the theore re ow: b c etc etc etc 3

4 We otice tht ( b)( c)( d)( ) = α ( b )( b c)( b d) ( b ) ( c )( c b)( c d) ( c ) ( d )( d b)( d c) ( d ) etc = β = γ = δ ( )( b)( c) ( y) = ζ ζ = ( )( b)( c) ( y) Keep i id tht we just did prtil frctio expsio of this s we do couple of preliiry clcultios We see tht α = ( b)( c) ( y)( ) = ( b)( c) ( y) ( ) = A A = b c Siilrly for,,etc β γ do the followig clcultio: Now, puttig this together with our prtil frctio expsio, we c ζ = ( )( b)( c) ( y) A B C Y = b c y b c y = α β γ υ Now coes the puch lie Fro this lst equtio we get Tkig = 0 gives the desired result b c = 0 α β γ ζ 4

5 There re probbly y other wys to prove this result, but probbly o other wy hs such uexpected d surprise edig, d still uses oly 8 th cetury ethods Oce gi, Euler shows why he ws the gretest of his cetury Refereces: [E0] [E366] Euler, Leohrd, Itroductio to the Alysis of the Ifiite, trslted by Joh D Blto, Spriger, New York, 988 (v ), 990 (v ) Euler, Leohrd, Istitutioes clculi itegrlis, volue secudu, St Petersburg, 769, reprited i Oper Oi Series I vol Thks to Rob Brdley for his help with this colu Ed Sdifer (SdiferE@wcsuedu) is Professor of Mthetics t Wester Coecticut Stte Uiversity i Dbury, CT He is vid rtho ruer, with 3 Bosto Mrthos o his shoes, d he is Secretry of The Euler Society (wwweulersocietyorg) A ew issue of How Euler Did It ppers ech oth Copyright 005 Ed Sdifer 5