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Two square numbers whose sum, or whose difference, is also a square may be found from the equation

(2mn)2 + (n? m)? =(n? + m).

To find the values of x which satisfy the equation

a +bx+cx'=y.

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[1] Suppose c# f* ; assume (a+bx+82.0?)*=fx +

m-na

n'b 2mnf [2] Suppose a ¥ $2; assume (f2+bx+c«2)*=f+

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n

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[4] Suppose a +bx+ca? =(f+gw).(h+kx);

{U+8+) (4 + ka)}'U+gx), then

assume

fm*- hn?

kné - gm In this case be — 4ac must be a square.

(E. 38–62; B. 167–70.)

(59.) Reduction of the general equation.

The most general form of a quadratic equation is

a + bxy + cy® + dx + ey+f=0, which by assuming (by+d)* — 4a(cyo +ey+f)=t,

6* 4ac=A, bd - 2ae=g, d4af=h;

and Ay+g=u, go- Ah=B; may always be reduced to the form

u - At=B. (1) from the solution of which the values of xc and y may be

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In these values of x and y, t and u may be either positive or negative, integral or fractional; in its most general form, the equation (1), by substituting and , for u and t, becomes

mu? - Ay=Bx?

In all cases in which this equation is possible, it transformed into another of the form

may

be

X;? –Y=cm, (G. A. 26, 7; B. 172, 3; Leg. 15, 16.)

(60.) Reduction of the equation 2 2 - ay'=bxo.
The following conditions may be supposed to exist :

[1] x, y, and %, are prime to each other.
[2] y is prime to b, and x to a.

[3] a, and b, have no quadratic factor.
[4] a, and b, are both positive.

[5] b>a.
Assume

x=ny-bye n being such that nạ - aceb: let the quotient be b, k*, where by contains no quadratic factor; then

b, k* y* 2nyyı + by,' =x?. Multiply this equation by b, k”, and assume b, köy-ny, = x1

and kéz-=x;, then

x? - ay,' =b,%,?, which is similar to the original equation, except that b, < $6. If b, is equal to unity, or to any square, the required transformation is effected: the values of a, y, and %, are

nx, + n'yı

b, ko

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-by

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If b, is not a square, and > a, the same process may be repeated, and we obtain

X ay," =b,za. In which equation b, <<b,: by pursuing this method we must at length arrive at an equation in which bm is either 1, a square, or <a. In the latter case, by transposing, putting c for bm, and neglecting the subscript indices of x, y, and x, we obtain

- cx=ay. Proceeding as before, this may be reduced to another equation

x – czo=amy, in which an < c. Then similarly putting e for am, we have

x? - ey' = cx

By this process, the coefficient of either y or z must be reduced to a square, or to unity: the indeterminate quantities in the equation last obtained may be expressed in terms of X, y, and %, by successive substitution by means of the equations (1). The equation is thus reduced to

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&c.

&c. in which k”, 1°, &c. are the greatest integral squares in the

na. quotients

&c. b (Lagr. Mem. Berl. 1767; B. 174–6; Leg. 15–22; E. Add. 52.)

n?

a

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(62.) Solution of the equation x - ayo = +1.

Let Va be expressed by a continued fraction, (Art. 31.) and let

converging fraction corresponding to the

Am

be any

В,

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quotient 2e; the above equation will be satisfied by the values

y=B If the period consists of an even number of quotients, the equation que ayč=1 will be satisfied by every converging fraction corresponding to a complete period, and w? ayʻ=-1 will be impossible.

If the period consists of an odd number of quotients, the equations xay = -1, and we ayé = 1, will be alternately satisfied by the converging fractions corresponding to complete periods.

(B. 150; E. 96—111, Add. 37.) Having given one solution of the equation

w? ayo = +1, to determine the general values of w and y.

X? dye =1, and på - aq*=1;
={{p+qva" +p-qVal" }

y=xVa{P+9 Val"-p-9Val"}.
[2] 2 ? ayo =1, and p? - aq'= -1;
x = = {{p+qva]" +p-qvae"},

Va{P+gvasom-p-qvae".
[3] po? - ay= -1, and på - aq' = -1;
x={{p+qva

+Valem-:+p-qValem-1} ,
y

2 Va{p+gValm-?-p-9V.am}. (B. 180.)

y =

L

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