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

(2mn)2 + (n2 — m2)2 = (n2 + m2)2.

To find the values of a which satisfy the equation

a + bx + cx2=y3.

[1] Suppose cf2; assume (a + bx +f2x2)* =ƒœ+

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[2] Suppose a ‡ƒ2; assume (ƒ2 + bx + cœ2)*=f+

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[3] Suppose a=0; assume (bx+cx2)*:

m x

mx =~ n

then

n

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[4] Suppose a + bx + cx2= (f+gx).(h+kx);

assume

then

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In this case b2-4ac 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

ax2 + bxy+cy2+dx+ey+f=0,

which by assuming (by + d)2 - 4a (cy2+ey+f)=t, b2-4ac4, bd-2ae=g, d2-4af=h;

and Ay+g=u, g2-Ah=B ;

may always be reduced to the form

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from the solution of which the values of x 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 2, for u and t, becomes

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

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(60.) Reduction of the equation x2 — ay2=bx2.

The following conditions may be supposed to exist:

[1] x, y, and x, are prime to each other.

[2] y is prime to b, and ≈ to a.

[3] a, and b, have no quadratic factor.

[4] a, and b, are both positive.

[5] b>a.

Assume

x=ny-by1,

19

n being such that n2ab: let the quotient be b1k, where b1 contains no quadratic factor; then

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Multiply this equation by b, k, and assume

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which is similar to the original equation, except that b1 <b. If b1 is equal to unity, or to any square, the required transformation is effected: the values of x, y, and ≈, are

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If b1 is not a square, and >a, the same process may repeated, and we obtain

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m

be

In which equation b<b1: by pursuing this method we must at length arrive at an equation in which bis 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 ≈, we obtain x2 — cx2=ay2.

1

Proceeding as before, this may be reduced to another

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

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in which k2, l2, &c. are the greatest integral squares in the quotients

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-

a n2.

C

,

&c.

b

α

(Lagr. Mem. Berl. 1767; B. 174-6; Leg. 15-22; E. Add. 52.)

(61.) Solution of the equation x2 — y2 = ax2.

Let aa1.ɑ29 and = = 81.829 then

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(62.) Solution of the equation x2-ay2=±1.

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

and let

Am
Bm

be any converging fraction corresponding to the

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

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If the period consists of an even number of quotients, the equation x2 - ay2 = 1 will be satisfied by every converging fraction corresponding to a complete period, and x2 — ay2 ——1 will be impossible.

If the period consists of an odd number of quotients, the equations x2-ay2-1, and a-ay-1,

2

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

x2 — ay2 = ±1,

to determine the general values of a and y.

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{p+qvam1-p-qvam-1}. (B. 180.)

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