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. [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+ n [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 x® + bxy + cy® + dx + ey+f=0, which by assuming (by+d)* — 4a(cyo +ey+f)=t, 6* — 4ac=A, bd - 2ae=g, d’ — 4af=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 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. [1] x, y, and %, are prime to each other. [3] a, and b, have no quadratic factor. [5] b>a. 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 -by 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 &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 (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, 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 x— ay = -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; y=xVa{P+9 Val"-p-9Val"}. Va{P+gvasom-p-qvae". +Valem-:+p-qValem-1} , 2 Va{p+gValm-?-p-9V.am}. (B. 180.) y = L |