INDETERMINATE ANALYSIS. 



■ ;es, as in the prefent cafe, tlie number of folutions i? 



limited, and in many cafes the cquati;)n is innpofiible ; 



however be demonilrated, that the equation always 



of at leaft one folution, \{ c > a b — a — i, a andi 



lime to each other ; and it is propofed in the prefent 



ion to afccrtain the exadl number of poflib'e iohi- 



t'.j.ja that any equation of this kind admits of in integer 



numbers, and to point out more accurately the limits of pofii- 



bility. 



The folution of the equation ax ->r iy =■ c, depends, like 

 that in the foregoing propofition, upon the equation ap — 

 A y = ± I , though its connettion with it is not fo readily per- 

 ceived. 



Let ap — bq— I ; then we have alfo 

 a.sp — b.cq=^c-, 

 and it is eviJent thnt we (hall have tlie fame refult if we make 



.X =: cp — mb ; and y := s j — 1:1 a ; 

 for this alfo gives 



a [cp - mb) - b {eg - ma) = c : 

 affuming, therefore, for m fuch a value, that c 7 — m n may 

 become negative, while c^ — ot 3 remains pofitive, we fnall 

 have 



a(cp—mb)+b{ma — cq)r=c; 



and confequently x = cp — mb; and y — ma — c q ; 

 .but if m cannot be fo taken that cq — ma may be negative, 

 vhile c p — mb remains pofitive ; it is a proof that the pro- 

 j)ofed equation is impoluble in integer numbers. And on the 

 contrary, the equation will always admit of as many integral 

 folutions as there may be different values given to m, fuch 

 that the above conditions may obtain. 



And hence we are enabled to determine a priori the num- 

 ber of folutions that any propofed equation of the above 

 f jrm admits of ; for fince we mull have c p > mb, and c q < 

 ma, the number of folutions will always beexprefled by the 



greatcfl integer contained in the exprefGon ( f- — ~)' 



as is evident, becanfe tn muft be lefs than the fird of thofe 

 fraiftioDS, and great. .- liian the fecond, and therefore, the dif- 

 ference between tlie integral part of thefe fractions wi 1 

 exprefs the number of different values of m ; except when 



cefGvely 9 to the values of j', and fubtrafting ij from thofe 

 of .V ; thus 

 .r=2ij, 202, 189, 176, 163, 150, 137 &c. 

 y = 5' H. 23, 32, 41, 50, 59 &c. 

 that is ; 



9 . 215 4- 13 • 5 = 20CO 



9 . 202 -I- 13 . 14= 2000 



9 . 189 f ij . 23 := 2000 



9 . 176 + 13 . 32 = 2000 



&c. &c. &c. 



Ex. 2.— Given the equation 1 1 « -t- 13J' = 190, to find 



the number of folutions, and the values of x zndy. 



Firll in the equation 11^ — 13 7 = I, we have /i = 6, and 

 J =: J ; therefore 



is a complete intege 



i'hich is the i 



,we mult con 



and rejeft it, but not-, the reaf 



for which is obvious. 



£s. 1. — Required the values of .r and y, in the equation 

 9* + 13/= 2000, 

 and the number of pofPible folutions in integers. 



Firll, in the equation gp— ISq = l> we have at orce 

 f=^, and q:zz 2.; therefore the number of folutions will be 

 cxpreffed by 

 2000 X 5 

 13 

 And thefe are readily obtair.ed from the fbrmulx 



J^Jl = 9 Hi — 4000 



2000 X 2 _ 



444 



{::■ 



and 



p - m 



ooco —13m; Ly — 9 



in which, affuming m = 445, 446, &c ; in order that 9 nj 

 > 4000, we fhaii have the following folutions, each of 

 which is deduced from the preceding one, by adding fuc- 



whence it follows that the equation admits of only one in- 

 tegral folution, and this is obtained from the formuhc 

 f .V = cp - m b, or ^^j f^ = "' '' - ^ 7- "^ 

 J^.x:=I90.6 — IjW; t_y=llm— 190.J 



where, by taking m = 87, in order that ma — cq may be 

 pofitive, we have .v = 9, and ji ^ 7, which gives 

 1 1 . 9 -f 13.7= 1 90, as required. 

 Prop. IV. 



To find the values of x,y, and s, and the number of in*- 

 tegral folutions of any equation of the form 

 ax+ by + cz= .A 

 In the firft place we may obferve, that if any one or more 

 of the co-efficients a, b, or c, be negative, the number of 

 anfwers is indefinite. For let b be negative, then the equa- 

 tion may be put under the form 



ax + cz = by + di- 

 in wl.ieh, by means of the indeterminate^', ati infinite num- 

 ber of values may be given to the fecond fide of the etiua- 

 tion ; and confequently alfo to x and y. We need, there- 

 fore, only confider equations of the form above given, in 

 which the quantities are all connofted with the fign +. 



Now in tliis equation, as in thofe of the two foregoing 

 pmpohlions, if a, b, and c, have each a common divifor, 

 which d has not, it becomes impoflible ; but if only two of 

 them, as a and b, have a common meafure, the equation ia . 

 llill poffible, as we fhallfee in what follows. 

 The folution of the equation, 



ax + by-\-c%r=.(i, 

 is refolved by means of the equation 



ap— bq = ± i». 

 as we have feen is the cafe in the preceding example?. 

 For let one of the three terms, as e z, be tranfpofed to • 

 the other fide of the equation ; then we have 



0,1- 4- iji = d— c z, 

 in which the values of x. and y, determined as in the lall 

 propofition, will be 



X — {d — c s)j^ — mb; and j' = ;na — [d — c %) q, 

 that is, by only fubftituting d — c a inftead of c, which 

 is the only refpcft in which this equation dilFers from that 

 in tlie foregoing problem : and here the only limits to be 

 obfervxd are, 



ift, r z < rf; 2d, m * < (J — f =) / ; 3d, m a >((/ — f al y 

 by attending to ^vhich, all the poifible values of x and ji 



may 



