Solution #25

First, note that 104729 is prime. For brevity, denote it as p.

Then we want x and y such that 2/p = 1/x + 1/y, or 2xy = p(y + x).

From this, it follows that p is a factor of 2, of x, or of y. It is not a factor of 2. Assume first that it is a factor of x. Then x = pa, for some positive integer a.

Thus 2apy = p(y + ap), or 2ay = y + ap, or (2a-1)y = ap. But the greatest common factor of a and 2a-1 is 1, so a must be a factor of y. Thus y = ab, for some positive integer b.

Thus (2a-1)ab = ap, or (2a-1)b = p. Thus either 2a-1 = p, or b=p. Suppose first that b=p. Then 2a-1 = 1, and a = 1. Then x = y = p.

Suppose instead that 2a-1 = p. Then b = 1. Then a = (p+1)/2, so x = p(p+1)/2, and y = (p+1)/2. In our particular case, x = 5484134085, and y = 52365.

If, instead, we assume p is a factor of y, we get the same options reversed. Thus the possible ordered pairs are (104729,104729), (5484134085,52365), and (52365,5484134085).