!2^8 L. EULERI OPERA POSTHUMA. Arithmetica. 



aa pp H-p — mm -+- 1. <> ■, 



cc m^pp — ^imm^mm — l)*p-t- (mm -;- 1)' 



hujusmodi aulem binae formulae supra sunt resolutae. Ita si sumatur m = 2, ut sit q=h-p — 3, hinc reperitur 



figoQ 



P= CA ^ r 'j - H*^ autem solutiones diversae erunt ab iis, quas prior solutio suppeditaverat. Ceterum hic 

 statim ab initio scribi debuisset mm — 1 loco nn. 



SoLUTio GENERALiOR. Loco » et o scribatur — et — et formula resolvenda erit %, — —. Jam ponatur 



r r q^ — prr *■ 



p = 1 -1- a^ , q=.\-\-pz et r=.\-\~yz et habebimus 



3a _ p-2y-i- (3«a - 2/3}- - 77) 2 -h (a^ - /J77) «z 



D. 



3p — a — 2y -*- (3^/3 — la-j -ri)&^ (/J^ — ayy) %z ' 

 Hic igitur tantum opus est, ut fiat 



t*Uu.,^k=:« 3a-^-2y ^ /r o..^ (3«-/3)gg-(3j3-a)^ 



tinairn vtqnif»^ ' 3/3 — a — 27 sffli ' '^ gg — ff 



Hoc modo prodit ^=1433 el -'=473. Veluti si a = 2 et /?=1, fit y — ^g^-^ — JggL ^|. Erffo si 



1 1 



^=1 et f=2 eril 2/= — — , ideoque y= — — , unde fit 



00 



3.64-4-433Z-4-28722 



3.16-t-132z-+-34z« ~°' 



Ceterum hic nil impedit, quominus sumatur vel a = 0, vel ,5 = 0, vel y = 0; tantum sumi non debet (i = a. 



^ . . ,. . . 1 ^- •. 'i /-, r. .4 -♦- JB« -f- Cz« , . A 



Quovis autem casu simplicissima solutio ita repentur : Lum nat — = n , in qua aequatione — per 



Ci — r— OZ I CZZ Q 



A . 



hypothesin =n, ponatur hoc □= — , indeque prodit z. Sequens solutio imprimis est memorabilis, sumendo 



p =: [i ~t- nn) z , 5 = 1-4-2, ac per artificium modo memoratum reperitur 



(nn-Hl) (n*— 3nn-i- 1) , „, n^ — 2n* -♦- nn -h 1 • n^ -»- n* — 2nn -i- 1 

 ^= 3;^^ ' ""•^^fi^ P= Siii. "* ^= 3^i ' 



unde pro solutione formulae ah{(ia-\-hh)-=cd[cc-\-dd) statim habetur «=3»^ h = n^ — 2w*-*- wn-i-1, c=3nn, 



rf=n (n^H-w^— 2nn-H 1), quandoquidem posueramus h = cp, d = aq, hinc autem colligitur — =n^ hineque 



— = n^. Quodsi jam pro casu simplicissimo sumatur n = 2, fit a = 96, 6=37, c=12, rf=146 hincque erit 

 c 



a6 = 2^3.37, crf = 2^.3.73, aa -1-66 = 5.29.73, cc-i-rfrf= 4.5.29.37. 

 Theorema. Ex qualibet resolutione aequationis ah{aa-\-hh)=cd[cc-\-dd) semper alia solutio deduci potest. 



Demonstratio. Quia ah [aa -\-hb) :== cd [cc -i~ dd) , erit (a-H*)*— (a — 6)*= (c-f-rf)*— (c — rf)* hinc 



(aH_t)4_(c-4-rf)4— (a_6)4_(c_-rf)% geu * 



)iiK)'iq (Htm 'ttufl u)R(H>q 



{a-t-h-\-c-i-d)[a--*-b-'C — d)[D-+'n) = [a — b-\-c — d)[a — b-'C~t-d){a-i-n). 



Quamobrem si ponamus a'=a-i-b-\-c-\-d et h' = a-\-b — c — d; dein etiam 



c=a~\-b — c — d et d':=a — h — c-\-d 



eni ab^aa-^-b^b^^^cd' [cc-\-d'd'). Quia igitur erata=291, 6 = 25, c = 75, rf= 193, erit a'=584, 

 6'= 48, c'=384, /=148, qui p?r 4 depressi dant .„„,«.,1,« ,«Weil 



»«-f.n>t-*«€«'=**6' *'=12, c'=96, rf = 37, 

 quae est solutio posterior minima. 



