A is N, and cannot be resolved into factors, each of which is less than N, we have no general method of solution for integral values; in fact, the equation will not always admit of such values, although there may be fractional ones that will obtain, which is not the case if A</N, or resolvible into factors that are N. We must, therefore, in this case, employ a different method of solution; that is, we must find the values of t and u in the equation by art. 176, and then, dividing the whole by 2, we have and, calling this the known case, we shall have the general values of x and y, by means of the equation p2 - Nq2 = ± 1, as in the foregoing article; that is, Sx=pm± Nqn, y=pn ±pm; only in this, the general values may be fractional instead of being integral, as in the former case. Hence it appears, that in all cases when one solution is given, as many others may be deduced from it as we please; and when the given case is integral, all the other solutions will also be integral; and when the first is fractional, all the other de pendent solutions will be fractional likewise. Cor. The methods that have been explained, in the preceding proposition, for finding the general values of x and y in equations of the form x2 — Ny = A, are equally applicable to equations of the form x2 - Ny2 = Az3, as is evident; because this equation being multiplied by p2 - Nq2 = 1, will leave the second side of it the same as at first. PRACTICAL EXAMPLES. 1. Find the least integral values of x and y in the indeterminate equation x2 - 5y2 = 1. Ans. x=9; y=4. 2. Find the integral values of x, y, and z, in the indeterminate equation x2 — 5y2 = 13%2, or prove that there are no such values. Ans. Impossible. 3. Required to ascertain the possibility or impossibility of the equation 5x2 - 7y2 = 11%2. Ans. Impossible. 4. Find the least integral values of x and y in the indeterminate equation x2-7y2 = 1, and also in the equation x2-7y=-1, if the latter be possible. Ans. x=8, y=3, 1st equation. Impossible, 2d equation. 5. Find the two least integral values of r and y in the equation x2-13y2 = 1. x=649, 842431; y= 180, 233640. 6. Find the least values of x and y in the equa¬ 7. Required the least integral square that, when multiplied by 113, shall exceed another integral square by unity. Ans. x=1204353, y= 99296. 8. Required the least values of x and y in the equation 79x2 - 101y2 = 1. 396 CHAP. IV. On the Solution of Indeterminate Equations of the Third Degree, and those of Higher Dimensions. PROP. I. 184. To find rational values of x in the equation ax3 + bx2 + cx+d=z2. It is only under one partial condition of the absolute term d, that this equation admits of a direct solution, that is, when d is a complete square, as d=ƒ3, in which case the equation becomes 2 ax2 + bx2 + cx+f2 = x2; and, when this condition has not place, we have no other method of proceeding but by trial; and even when it has, we can find but one solution at a time, which is obtained in the following manner: 1st Method. Assume c ax3 + bx2 + cx +ƒ3 =ƒ3 + cx + 4ƒ3ïx2, or Ex. 1. Required the value of r in the equation x2 + x2+3x+1 = x2. Here a=1, b=1, c=3, and f=1; therefore, 23 which value, substituted for æ, gives 2=(~~) quired. -), as re 2d Method of finding the value of x in the equation Assume ax + bx2 + cx +ƒ2 = x2. z=f+gx+hx2; then, by squaring, we have a.x3 + bx2 + cx+f2 = ƒ2 + 2ƒgx + (x2 + 2ƒh)x2 + 2hgx' + h*x*. Which values, being substituted for h and the above expression for x, give x= 8af4bcf+c x 8fs. |