Solutions to Problems; Chapter (^8 373)
Try w = 1. Then we wish to find v and z such that 12v2 - 23 = ,r2. Two
obvious solutions are (v, %) = (2,5), (4,13). We can find as many more as
we wish by the following device:
If t2 - 12s2 = 1 and z2 - 12v2 = -23, then
(t + sdE)(t - s&q = 1
(z + vfi)(z - vfi) = -23
j (t + sfi)(z + vm)(t - sfi)(z - vfi) = -23
- [(t% + 12sv) + (tv + S%)dis][(tZ + 12sv)
- (tv + sz)fi] = -23
=s (tr + 12s~)~ - 12(tv + ~2)~ = -23.
In particular, t2 - 12s2 = 1 is satisfied by (t, s) = (7,2), so that if (v, %) sat-
isfies 12v2-23 = z2, then we can obtain another solution (7v+2z, 7z+24v).
Hence (v, z) = (2,5) gā Ives rise, successively to (24, 83), (334, 1157),.. ., and
tv, 2) = (4913) gā Ives rise successively to (54, 187), (752,2605),....
We can try other values of w to obtain solutions. For example, w = 2
requires making 4(3v2 - 23) a square, which will occur if v = 3. Here are
some possibilities:u x such that 3x2 - 5x + 4 = u2(^2 51390)
(^4) 3, -413
(^24) 44/3, - 13
54 32, -91/3
334 58113, -192
752 435, -1300/3
312 l/2,7/6
- Suppose, if possible, that there is a nontrivial rational solution. Be-
cause of the homogeneity of the left side, there must be a nontrivial integer
solution. For such a solution, xy% # 0 (otherwise, either 3 or 9 must be the
cube of a rational number). Let (x, y, 2) = (u, v, w) be a nontrivial solution
which minimizes 1x1 + IyI + 1~1. Clearly, u = 3t for some integer t. Then
(x, y, z) = (v, w, t) is also a solution and [VI + [WI + ItI < IzI[+ Iv1 + Iwl,
which contradicts the minimal property of (u, v, w). The result follows. - (a) The equation is equivalent to x + y = 4 and the general solution is
givenby(x,y)=(2-t,2+t)fortEZ.
(b) The equation is equivalent to x + (y - 6)x + y(y - 6) = 0. The
discriminant of the quadratic in x is equal to -3(y - S)(y + 2), and this is
square only if y = -2, 0, 4, 6. Hence the solutions (x, y) = (4,-2), (6, 0),
(-2,4), (0~6).