a4 
DIOPHANTINE, 
“made equal to the greater fquare, or the fquare of their 
$ difference to the lefs. 
This problem differs from the laft in having the 
ptopokd fquares limited in form or magnitude. 
Examples. 
OF the refelution of a a icate equality. 
1. Find fuch’a value of x, that x + 1, aad x 
both be fquares. The ace is2. Factors 1 aad 2,;$fum 
, 2 a. £ and 2, + the two ial therefore x = 3, 
. Find «x when x? + x, and » oo eal 
fas mis 2x*, we are to find two ae Sie _fum 
x 49 x tox 
Bue a =—. i ne 
23 a “3 25 
x 2 
=, 24K = 25 5,and x = 29, 
a5 24 
= = hypothenule, and p = double produ& of 
va Se c FE pang cP — pare f{quares (per 23.), confequently 
s cs? — p z*vare fquares. We have only 
to ‘nd fuch a value of z, that c?z? may be = the faaare of 
pr. 
ce 
c 
EE e 
== x = the number required, i. e. the {quare of any 
op 
hypothenufe divided by the double produ of the fides 
sy ees 
5 2 = 
and £2” 
gives x 
a ; 6 
As 3, 4, 5, gives = = x, 5, 12, 13; gives —, &e, 
3. Make 2x7 + 31, and 4x 41, {quares, | while x has 
the fame are in both. The difference is 2 x* — 4 x, the 
faGors 2% and x — 2, the difference ~ + 1. Put the 
x 2 x? 
lefs fquare 4% -- 1 7 + | =— +x + 1, or 
16% = x* + 4x, or x = 12, and the propofed expreflions 
are 289 = 172 and 4g. 
4. Find fuch a value cf x, as may make 10 — 2 x, and 
TO — x* (qnares. Diff. x? —~ 2x. FaGors « andx — 2. 
The § diff. = 1 = lefs fquare = 10 — x,or x = 3. 
Or, § fum = x — 1, and x? 4-1 = 10 — 24, x’ 
== 0; 416 x = 3; 
5. Tomakeax +c ab x + ape 2s where 4 
is greater than 4, and ne iene cither ++ o 
Rule, 
Let 9g = vary Mult. the diff. of the given 
{quares by g, and fubtra&t the prod. from the lefs f{quare. 
Call the remainder r. Then if ¢, or —4, be a fquare, the 
cuplicate equality is refolvable: thus, find two f{quares m* and 
n*, tte diff. of which may be to the excefs of the lefs above r 
asi:9. Whenrisafquare,y + Vr ? mutt be made equal 
to n? = bx + d, but when —# is a {quare, n® muft be 
made equaltodx + d. 
—Meke 8x + 4,and 6x* + 4 both {quares; their 
difference is 2» and g = 3, produ@ 6x, andr =4a 
Here y + 2\? es YP tay + 4, orn — 
ea and m= y? + 4y 
3 
{quare. 
4=Pr4y- 
+4+25U—8 
A 
2 + 4; to avoid frac- 
+ 
= 
tions ay by 9, then 9 m* =z 12 5% + 489 + 36 
39° +12 y +9 to be made a fquare: y =3 
45 
or 2 
4 
5... eve = 49 = 6x +4, and* = 
64. 
~-S8x +42 
Pros. XII. 
26. To find two numbers, A and B, fuch that A + B, 
A? + B, and B?+ A, may be all fquares. Let x= A, 
36 B, 
Then A? + B= anid is a fquare. 
But A? + B= 3x7 4+ x 
And B+ A= : xt+ x vate to be fquares. 
The difference 4 is 9 “3 x’. Faétors 3 x* and 3 x°—1, &c. 
I — _—- \ 
em os he oe ===B. 
Pros. XIII. 
27. To find two numbers, A and B, fuch that, A +1, 
B+, A + B+ 1, and a B + 1, may be all fquares. 
1. Let x? +2, and B= x? — 2, and thetwo firft 
conditions are ¢ fatinfed bet z2x°-+ r,and4x+ 1, arealfo 
to Pe quares ; De al 168, B = 120. 
. Let nA+B+1 
=x + 6x 47S =a {quare is the cid cae, fup- 
pole it =2"; then2*— y?=9x°+ 6x, Factors x and 
9 + 33 or y? = 162° + 24. + 9 3"or 
- Bat B—A+1=7%°+ 18% 
+ 9 is alfo to be a a fquare, beatae = 3x—3P >= 9x07— 
18s + 93 or 2x oe 18 .° hee , B= 
624; B A = 86. B ay ee each off ‘which 
+ 1 is the fquare of . 75,93, and sr, 
Pros. XIV. 
28. To find three numbers fuch, that if the fquare 
of each a _ to the fum of the two, the three fuma 
may be {fq 
+ 2-41. if x, 2%, and 1, denote 
ee ae namie. the firk condition - fatisfied. 
But 
+o--1, and 3x%+ 1, are anil to be slay oe 
4 rence | is4x*—2x. Factors, and 4x » 3 dif. 
3x Ox 
cuadinet Sh Semele t, ie ott S— i oe oe Oe a — ya Pe 
5 4 3 9 4%; 3 
16 ; 
2" 3 and 1, are the numbers required. 
Pros. XV. 
29. To find two numbers, A and B, fuch that AB -+- A, 
AB -+ B, an nd AB+ A+ B, may be all {quares. 
et x= A, x*= == a fquare; then x* — sx 
= AB, an ne and 2x—1=—A But . 
B+B= dAB+A+ B= »? emis 
be made iquares. Diff is x. Faétors 2 and 4, 2 diff. 
x— 4, or x? 48 — iI, ora=t I+ ss Zand 
a = A, and B=x— 1 = 2. 
Scholium. 30. To find the proper fa€tors, into which 
the difference of any aig aes {quare ought to be reduced. 
uiée Xut % = one of the factors, and the given differ. 
ence tae ag 2 == the other 3 then by making the {quare 
of the § fum or ¢ difference equal to the egal or lefs o 
the meopoled {quares, find | fome fuitable value o 
4 
“Examples: 
