DIOPHANTINE. 
be made a {quare, fuppole 2 + 267? = 44 4078? + att, 
B 
4. 
or27=4P 4+ ab, 2a — abt = 46, and a? = — 
— J 
z= a {quare. If 6 = 1, then a=a ust oae 
OF oe. 49 and yee 
16 16 16 4 
8. Find fuch a value of v as may make 2 — 9 rational, 
v is lefs than 1 
Afflume v =f —a, or vt = 1—4a4+6a°~— 443 4 at, 
then 2— vt = 1+ 4a— 60% 4+ 44° — at = fuppofe 
rf2a— sal’ I+ 44—62a@ — 20a? + 25%; or 
12 
25a—200—= 4a'—at; 264= 24,anda=— ..¥ 
I 
575 22 -—1 —I 239 
28561. 169° 
9. Make 2x? + 2x + 44 rational fee when « is not 
I 
— 
I 
=I-ag=—, and /2—4' 
13 
8 
Affume 2x? box t4m2+a2ae?% = 44 Sax 
x? 
, . 2 2 eee — 
4Ox WP +I= +20@x3;%°—2a@x=4Aa I; 
— 
— 
—~2@ —tI,andx=@4 warren, 
To make at + 4a — 1a f{quare, let =4a-—1, ora= 
&+408+6P +4641 
, then at= 
256 
_ bt +4R 468 4 2606+1 
» orat+ 4a 
to be made a {quare, 
256 
P I+ Je b? _b'—260b3 + 168985" + 2605+ z 
f{uppofe i it—= 
then 4 3 + oe 
— 1305; 132b= 
4289 
264 
ea 
Find fuch values of » and y, that «*+ ay?= 3 
ay ie sae val, 
Affum 
250 
16898 8? — 260 53, or 28 wre = 8449 
__ 4223 +1 
8446, and 6 = 56 
4 
294396964 = 
» x ozg28, and 2x? + 2xn+4= 
, then 
my, Or x == 3b — 2 2b my + my" 
b? Sia de cor my? + ay? iS 
bm 
my + ay = 2bm, andy = But x? eas 
P 4am _ BPxa+m|)— 4am 
at mp 
2 * 2am’ + m4 b.a—m 
Me ae and x = + =b-—my 
a + m’\ a+ m 
; 26m? b.atn?—2m bam 
=o . 
er a +m 2? a+ m* 
Affome OF gee axn—x7 + x}, 
or « 2x? = 2. 
12. Make ; +6x—5x?+ 3% a fquare. 
i x + a? x7, 
make the fecond term vanifh from each fide, as well as the 
ih, 4a muft be = 6, ora = ©. Then 2 = 3x5 0 
= 2, and. the expreffion 
or, 4 = 3”, and # — 
-and leffened either 
ee 29 5 2s)" 45 5 J 235 
2°32 = 3 =555 0 
m 2025 
64 : 
13, Make a +hx% ten? +dx? + ex a fquare. 
Affumeit = 4+ mx fax? =a? +2amn +70 t x 
+2mnx +7°xt, Here 2am =b,orm= 
b 
—,and 2an 
22° + 
4 c 
m* = ¢,orn = 
2 
—m 
»thendx? + ext=2mnxi +n: 
a 
d-+ex=iwmn+nx; ex~—nx=2mn—d, and # 
2mm — nid 
— 
- Make a+texta ale ee one Pion Of =H, 
and a+ en*=s, Tao find other value 
+en*+ genty eee 
writings? for a + ent. niy+tOeny? + 4 
++ ey* is to be made a {quare, as the 13th. 
Prog. II. 
15. To find two numbers, x and y, fuch that x? + y _y may 
be a fquare, and x + y its root. We have x? eas = : «+ yf 
HSH f+ axy t+ Py =IwAH J+tTM 7 +2 oe dy=1 
— 2, where x may be any eee lefs than 3 4. "Sa ppofe 
xo 4,t a aa 20° + yomdgandx+y= 
= 
its root, if 
= = 38 + y= gy, and x + y = 4, &e. Ke. 
Pros. III. 
. To find two rane x and DD 
be a iquare, and x* +- cot 
Let Sy mas then shy a's, whencey = ot 
fuch that « + y may 
—xrnx— ex t+ wma I,and x= — 
erie 
Te ay oppoen = 250 = 8 ym 2, an x+ys 
5 
36 a g6 _ & 
a and x? + y = e ra 
Pros. IV. 
17. To find a number which being divided into any two 
parts, xand y, « may be always equal to He 
et am sd bz denote the required parte, thn aoe % 
eid ia edad z— —é,and 
a and oe = t, and 
= FORK Tp aes = j = one part, an 
bz = ag tae the other part, and both added together = 
a 
2 ; = 1, the number required. 
For let « and 1 — x be any two parts of 1, 
I—wemie «pt 
Then x? + 
x= I1— 26 + x, or, 2difx? + y 
+ x 
= + mee ey x—y,thenx-+y = 1. 
wo numbers, fuch that their {um being increafed 
i difference, or the d:fference 
of their fquares, the fums and remainders fhali be all 
fquares. 
Let 25 x? be the fum, and 24 x? the difference of the 
required numbers. Then 25 x? + 24 x" are evidently fquares. 
More- 
